In the sequel, I will transcript my notes from Shen’s talk.

**1. Introduction **

In Real Analysis, the classical Weierstrass function is

with .

Note that the Weierstrass functions have the form

where is a -periodic -function.

Weierstrass (1872) and Hardy (1916) were interested in because they are concrete examples of continuous but nowhere differentiable functions.

Remark 1The graph of tends to be a “fractal object” because is self-similar in the sense that

We will come back to this point later.

Remark 2is a -function for all . In fact, for all , we have

so that

whenever , i.e., .

The study of the graphs of as fractal sets started with the work of Besicovitch-Ursell in 1937.

Remark 3The Hausdorff dimension of the graph of a -function isIndeed, for each , the Hölder continuity condition

leads us to the “natural cover” of by the family of rectangles given byNevertheless, a direct calculation with the family

does notgive us an appropriate bound on . In fact, since for each , we have

Fortunately, we canfor . Because is arbitrary, we deduce that . Of course, this bound is certainly suboptimal for (because we know that anyway).refinethe covering by taking into account that each rectangle tends to be more vertical than horizontal (i.e., its height is usually larger than its width ). More precisely, we can divide each rectangle into squares, say

such that every square has diameter . In this way, we obtain a covering of such that

for . Since is arbitrary, we conclude the desired bound

A long-standing conjecture about the fractal geometry of is:

**Conjecture** (Mandelbrot 1977): The Hausdorff dimension of the graph of is

Remark 4In view of remarks 2 and 3, the whole point of Mandelbrot’s conjecture is to establish the lower bound

Remark 5The analog of Mandelbrot conjecture for the box and packing dimensions is known to be true: see, e.g., these papers here and here).

In a recent paper (see here), Shen proved the following result:

Theorem 1 (Shen)For any integer and for all , the Mandelbrot conjecture is true, i.e.,

Remark 6The techniques employed by Shen also allow him to show that given a -periodic, non-constant, function, and given integer, there exists such that

for all .

Remark 7A previous important result towards Mandelbrot’s conjecture was obtained by Barańsky-Barány-Romanowska (in 2014): they proved that for all integer, there exists such that

for all .

The remainder of this post is dedicated to give some ideas of Shen’s proof of Theorem 1 by discussing the *particular* case when and is large.

**2. Ledrappier’s dynamical approach **

If is an *integer*, then the self-similar function (cf. Remark 1) is also -periodic, i.e., for all . In particular, if is an integer, then is an invariant repeller for the endomorphism given by

This dynamical characterization of led Ledrappier to the following criterion for the validity of Mandelbrot’s conjecture when is an integer.

Denote by the alphabet . The unstable manifolds of through have slopes of the form

where , , , and

In this context, the push-forwards of the Bernoulli measure on (induced by the discrete measure assigning weight to each letter of the alphabet ) play the role of *conditional measures along vertical fibers* of the unique *Sinai-Ruelle-Bowen (SRB) measure* of the expanding endomorphism ,

where and . In plain terms, this means that

where is the unique -invariant probability measure which is absolutely continuous along unstable manifolds (see Tsujii’s paper).

As it was shown by Ledrappier in 1992, the fractal geometry of the conditional measures have important consequences for the fractal geometry of the graph :

Theorem 2 (Ledrappier)Suppose that for Lebesgue almost every the conditional measures have dimension , i.e.,

Then, the graph has Hausdorff dimension

Remark 8Very roughly speaking, the proof of Ledrappier theorem goes as follows. By Remark 4, it suffices to prove that . By Frostman lemma, we need to construct a Borel measure supported on such that

where . Finally, the main point is that the assumptions in Ledrappier theorem allow to prove that the measure given by the lift to of the Lebesgue measure on via the map satisfies

An interesting consequence of Ledrappier theorem and the equation 1 is the following criterion for Mandelbrot’s conjecture:

Corollary 3If is absolutely continuous with respect to the Lebesgue measure , then

*Proof:* By (1), the absolute continuity of implies that is absolutely continuous with respect to for Lebesgue almost every .

Since for almost every implies that for almost every , the desired corollary now follows from Ledrappier’s theorem.

**3. Tsujii’s theorem **

The relevance of Corollary 3 is explained by the fact that Tsujii found an explicit *transversality condition* implying the absolute continuity of .

More precisely, Tsujii firstly introduced the following definition:

Definition 4

- Given , and , we say that two infinite words are -transverse at if either
or

- Given , , and , we say that two finite words are -transverse at if , are -transverse at for all pairs of infinite words ; otherwise, we say that and are -tangent at ;
- ;
- .

Next, Tsujii proves the following result:

Theorem 5 (Tsujii)If there exists integer such that , then

Remark 9Intuitively, Tsujii’s theorem says the following. The transversality condition implies that the majority of strong unstable manifolds are mutually transverse, so that they almost fill a small neighborhood of some point (see the figure below extracted from this paper of Tsujii). Since the SRB measure is absolutely continuous along strong unstable manifolds, the fact that the ‘s almost fill implies that becomes “comparable” to the restriction of the Lebesgue measure to .

Remark 10In this setting, Barańsky-Barány-Romanowska obtained their main result by showing that, for adequate choices of the parameters and , one has . Indeed, once we know that , since , they can apply Tsujii’s theorem and Ledrappier’s theorem (or rather Corollary 3) to derive the validity of Mandelbrot’s conjecture for certain parameters and .

For the sake of exposition, we will give just a flavor of the proof of Theorem 1 by sketching the derivation of the following result:

Proposition 6Let . If and is sufficiently large, then

In particular, by Corollary 3 and Tsujii’s theorem, if and is sufficiently large, then Mandelbrot’s conjecture is valid, i.e.,

Remark 11The proof of Theorem 1 in full generality (i.e., for integer and ) requires the introduction of a modified version of Tsujii’s transversality condition: roughly speaking, Shen defines a function (inspired from Peter-Paul inequality) and he proves

- (a) a variant of Proposition 6: if integer and , then for some integer ;
- (b) a variant of Tsujii’s theorem: if for some integer , then .

See Sections 2, 3, 4 and 5 of Shen’s paper for more details.

We start the (sketch of) proof of Proposition 6 by recalling that the slopes of unstable manifolds are given by

for , , so that

Remark 12Since , the series defining converges faster than the series defining .

By studying the first term of the expansion of and (while treating the remaining terms as a “small error term”), it is possible to show that if , then

(cf. Lemma 3.2 in Shen’s paper).

Using these estimates, we can find an upper bound for as follows. Take with , and let be such that distinct elements listed in such a way that

for all , where .

From (3), we see that

for all .

Since

for large enough. Indeed, this happens because

- if ;
- if ;
- as , and as (here we used ).

By combining (4) and (5), we deduce that

for all .

Since , the previous estimate implies that

i.e.,

Thus, it follows from our assumptions (, large) that

This completes the (sketch of) proof of Proposition 6 (and our discussion of Shen’s talk).

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- The Third Palis-Balzan International Symposium on Dynamical Systems will be held at
*amphithéâtre Hermite*in Institut Henri Poincaré (Paris, France) next week. - The Workshop on Combinatorics, Number Theory and Dynamical Systems will be held at IMPA (Rio de Janeiro, Brazil) from August 24th to August 28th, 2015.

The Third Palis-Balzan International Symposium on Dynamical Systems closes the five-year long *Project Palis-Balzan – Dynamical Systems, Chaotic Behaviour-Uncertainty*, sponsored by the Balzan Foundation, related to the prestigious award conferred to Jacob Palis (and IMPA) by the Balzan Foundation in 2010.

A detailed description of the program and the titles and abstracts of talks of this conference can be found here and here.

The Workshop on Combinatorics, Number Theory and Dynamical Systems is the second edition of an event organized by C. Mauduit, C. G. Moreira, Y. Lima, J.-C. Yoccoz and myself back in 2013.

The full list of speakers for the 2015 edition of this workshop can be found here.

I guess that this is all I have to say for now (but you can look at their respective webpages for updated information). See you in Paris or Rio!

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Proposition 1Then, there exist , , and such that the iterates of the weighted transfer operator satisfyLet be an uniformly expanding Markov map on and let be a good roof function with exponential tails.

for all and with , .

Remark 1We use the same terminology from the previous post of this series.

Roughly speaking, the basic idea behind the exponential contraction property in Proposition 1 is that “*oscillations produce cancellations*”. In particular, the analysis of the “size” of is divided into two regimes:

- (A) If exhibits a high oscillation at scale (in the sense that ), then we will have a “cancelation” (significant reduction of the size of ) thanks to classical methods (Lasota-Yorke inequality);
- (B) If the oscillation of at scale is not high, then we will have a “cancelation” thanks to Dolgopyat’s mechanism, i.e., a combination of high oscillations of Birkhoff sums of the roof function (coming from the fact that is not a -coboundary) and the big phases , , of the terms in the formula defining .

In the remainder of this post, we will formalize this outline of proof of Proposition 1. More precisely, the next section contains a discussion of Lasota-Yorke inequality and the regime (A), and the last section is devoted to Dolgopyat’s cancelation mechanism and the regime (B).

**1. Lasota-Yorke inequality and regime (A) **

Before attacking Proposition 1, let us warm up with a digression on the spectral properties of the weighted transfer operators .

The usual transfer operator acts on the space of functions. This action has a simple isolated eigenvalue at . The eigenfunction associated to the eigenvalue and normalized so that is the density of the unique absolutely continuous invariant probability measure of . Furthermore, the essential spectral radius of is and has no eigenvalues of modulus except for . (See Aaronson’s book for more explanations.)

For close to , the operator is a small perturbation of . In particular, has an eigenfunction associated to its unique eigenvalue close to such that and converges to in the -topology as .

From now on, let us fix such that is well-defined and bounded away from zero for . (Here, is the constant appearing in the exponential tails condition for the roof function .)

From the technical point of view, it is convenient to “uniformize” this spectral picture by normalizing the operators (for with ) as follows:

The *normalized weighted transfer operator* satisfy and . In other words, if we replace by , we normalize both the eigenvalue and the eigenfunction to . Moreover, the proof of Proposition 1 can reduced to the analogous statement for the normalized operators:

Proposition 2Suppose that there exist , , , and such that

for all and with and .Then, the conclusion of Proposition 1 is valid.

The proof of this proposition is based on *Lasota-Yorke inequality* (cf. Lemmas 7.8 and 7.9 of AGY paper):

Lemma 3There exists a constant such that, for all and with , , we have

- for all ;
- if , then ;
- if , then ,

where is the expansion constant of .

Before proving this lemma, let us use it to show Proposition 2. Recall that the assumption of this proposition is that

for some fixed , , and for all and with , , and our task is to prove the analogous statement in Proposition 1 for .

The idea is very simple: from the spectral discussion above, it is not hard to see that we introduce a factor of the order of when replacing by ; since is close to , this factor does not significantly affect definite contraction on the size of provided by the hypothesis of Proposition 2. Let us now turn into the details.

Given , we write with . Since and is uniformly close to , the previous estimate for the normalized operator gives that

for all with , . By the last item of Lemma 3, it follows that

Because is close to for small, we can choose such that

we deduce from the previous estimate that

for all and with , . This completes the proof of Proposition 2 modulo Lemma 3.

** 1.1. Proof of Lasota-Yorke inequality **

Let us now prove Lemma 3. The first item follows from a computation similar to the proof of Lemma 8 of the previous post, and its half-page proof is given in Lemma 7.8 of AGY paper.

For the sake of convenience, we provide just a sketch of proof. We write

where is the set of inverse branches of , and (with ). By taking the derivative , we obtain five terms , , , , depending by differentiating , , , or .

The terms and are easy to deal with: the uniform bounds on and , and the contraction of inverse branches of imply that and . Thus, .

Similarly, the distortion bound (Renyi condition) on (see the previous post) implies that , so that .

Since and (because is a good roof function and the inverse branches of contract exponentially), we see that and, *a fortiori*, .

Finally, the exponential contraction of inverse branches of says that , so that . This proves the first item of Lemma 3.

The second item is an immediate consequence of the estimate and the first item just proved. Indeed, if , then

Closing the proof of Lemma 3, we observe that the third item is a direct corollary of the second item.

** 1.2. First part of Proposition 1: study of regime (A)**

By Proposition 2, one can establish Proposition 1 by showing a contraction property for the normalized weighted transfer operators .

As we already mentioned, the first step in this direction is the treatment of the regime (A) of functions displaying high oscillations via Lasota-Yorke inequality:

Lemma 4Let with and and suppose that exhibits a high oscillation at scale in the sense that (where is the constant from Lemma 3). Then,There exists such that any has the following property.

*Proof:* The facts that is normalized and has high oscillations imply that

Furthermore, the Lasota-Yorke inequality in the first item of Lemma 3 says that

Since and , we get that

for and sufficiently large (so that ).

Of course, we can iterate this lemma to establish the contraction property in Proposition 2 *while* the high oscillation property is not destroyed by :

Corollary 5Fix where is the integer provided by Lemma 4. Suppose that is a function on and is an integer such that the high oscillation property is not destroyed by the first iterates of , i.e.,

for all and for some with , . Then,

*Proof:* By the third item of Lemma 3, we have that

By iterating Lemma 4, it follows that

This proves the corollary.

Remark 2The proof of the corollary shows that a strong (pointwise/) form of cancellation occurs in the high oscillation case (regime (A)). As we are going to see in the next section, one has a much weaker () form of cancellation in the low oscillation case (regime (B)).

**2. Dolgopyat’s mechanism and regime (B) **

After our success in dealing with the regime (A) (cf. Corollary 5), let us analyze the regime (B) of functions whose oscillation is not high.

** 2.1. UNI condition **

As we told in the introduction, the cancellation mechanism in regime (B) originates from the oscillations of the Birkhoff sums of the roof function . Here, by “oscillations of Birkhoff sums” we mean the following:

Proposition 6Let be an uniformly expanding Markov map. Let be a function such that

Then, the following conditions are equivalent:(where is the set of inverse branches of .)

- is not a coboundary: it is not possible to write with which is constant on each and .
- Uniform non-integrability (UNI) condition (“oscillation of Birkhoff sums”): there exists a constant such that
for some arbitrarily large and some inverse branches of .

This proposition is (contained in) Proposition 7.4 of AGY paper and we refer to it for the one page proof of this result.

Remark 3The main point of this proposition is that a qualitative property (“not a coboundary”) in our definition of good roof function in the previous post turns out to be equivalent to a quantitative property (“definite oscillation of Birkhoff sums”).The nomenclature “uniform non-integrability” (UNI) comes from the fact that this is a (uniform) quantitative property issued from the fact that the suspension flow associated to and is not integrable (conjugated to a suspension flow with piecewise constant roof function) when is not a coboundary.

** 2.2. Dolgopyat’s cancellation mechanism **

Let us now use UNI condition to produce non-trivial cancellations in some regions of the phase space for functions with “low oscillations”. In other terms, we want to study , , for such that .

For technical reasons, we will keep track of *and* an appropriate -bound for :

Definition 7Given , we say that if is , is a function providing an adequate bound for in the sense that

- , and

for all .

Remark 4is a cone and we are going to show in Lemma 10 that sends “strictly inside itself” in a certain sense.

Moreover, we need to anticipate the fact in Remark 3 that the cancellations in regime (B) occur only at certain spots of the phase space. In particular, the following “localization tool” (cf. Lemma 7.12 in AGY paper) will be helpful in the subsequent cancellation discussion.

Lemma 8There exists an integer such that any has the following property. Let with , , let , and let with , . Suppose that, for all , we have

Then, .

This lemma is a consequence of the Lasota-Yorke inequality in Lemma 3 and its half-page proof can be found after the statement of Lemma 7.12 in AGY paper.

Remark 5We call this lemma a “localization tool” for the following reason. We think of as a bump function supported on disjoint intervals of size . Next, we use to “localize” anappropriatebound in a pair to the support of by considering . In this setting, the lemma says that if the iterate of this localization gives a bound to , then is actually anappropriatebound for (i.e., ).

**From now on**, we use UNI condition (cf. Proposition 6) to **fix** and inverse branches such that

where , and are the constants in Lemmas 3 and 8. (This is possible because .)

In this language, Dolgopyat’s cancellation mechanism can be stated as:

Lemma 9Then, for every interval , we can find a point such that one of the following possibilities holds:There exist a small constant and a large constant with the following property. Let with , , and .

- Cancellation of type : for all , we have
- Cancellation of type : for all , we have

In other words, this lemma says that, for each interval of of size of order , we can find a subinterval of size of the *same* order where the two terms of

associated to the inverse branches and fixed above exhibits a *significant cancellation* w.r.t. the trivial bound (i.e., one gets a factor of instead of ).

This lemma is exactly Lemma 7.13 in AGY paper and we provide a sketch of its proof in the sequel.

*Proof:* Let , resp. be a small, resp. large constant. Consider and . Our task is to find with so that the conclusion of the lemma is valid.

The argument is divided into **two cases**. In the **first** (easy) **case**, we assume that the appropriate bound for is not “tight”, i.e., there exists such that or . In this situation, we want to take advantage of the non-tightness (and the fact that does not have high oscillations [as ]) to get some cancellation in an interval of size centered at .

For this sake, up to exchanging the roles of and , we can assume that . Since , we know that . Thus, because is a contraction. By Gronwall’s inequality, it follows that . Therefore,

for all . By integrating this estimate, we see that

for all . Since (by “non-tightness” assumption), we obtain from the estimates above that, if is small enough, then

for all . This proves the lemma in this “non-tight” case.

In the **second case**, we assume that the bound for is “tight”, i.e., and for all . In this context, we want to find such that the complex numbers

have *opposite* phases.

For this sake, we put for and we denote by the difference of the phases (arguments) of the complex numbers

Using UNI condition (2), the fact that does not have high oscillations (i.e., ) and the “tightness” of the bound for , it is possible to prove with a short calculation that

where . (See page 192 of AGY paper.)

It follows that if is sufficiently large (e.g., ), then there exists such that and have opposite phases.

We set . By exchanging the roles of and if necessary, we can assume that . By exploiting the fact that does not have high oscillations and the tightness of the bound for (via Gronwall inequality), it is possible to show (through another short calculation, cf. page 193 in AGY paper) that if is sufficiently small, then for all one has

and

where and denote the phases of the complex numbers and . In other terms, for any in the interval ,

- the size of does not drop too much in comparison with the size of : except for a factor , one still has the same comparison from the case ;
- and have almost opposite phases.

On the other hand, an elementary trigonometry lemma (cf. Lemma 7.14 in AGY paper) says that two complex numbers and such that

and

verify the “cancellation estimate”

By applying this lemma to and for , we deduce that if , then

that is, one has a cancellation of type .

This completes the sketch of proof of Lemma 9.

** 2.3. Second part of Proposition 1: study of regime (B)**

By Proposition 2 and Corollary 5, one can establish Proposition 1 by proving a -contraction property in regime (B) for the normalized weighted transfer operators.

Evidently, the key tool to get this -contraction property is Dolgopyat’s cancellation lemma 9 because it says that, given any interval of size , a non-trivial amount of cancellation for must happen inside a subinterval of comparable size . In particular, for the regime (B), even though we do *not* get cancellation *everywhere* in phase space, we still have cancellation in large chunks of the phase space. Thus, it is reasonable to expect a contraction property for in the norm (but not for the norm) in the regime (B).

Let us now try to formalize this heuristic argument. Fix such that where and are the constants in Lemma 9.

Lemma 10There exists and with the following property. Let with , , and let . Then, there exists such that and

In other words, this lemma says that for any in regime (B) (i.e., ), we can find an appropriate bound for whose -norm got contracted by a definite factor in comparison with the -norm of an appropriate -bound for .

This lemma is exactly Lemma 7.15 in AGY paper. For the sake of convenience, let us now sketch its proof.

*Proof:* We take a maximal set of points such that the intervals are pairwise disjoint and compactly contained in . Note that the intervals cover .

By Dolgopyat’s cancellation lemma 9, we know that each contains a subinterval such that a cancellation of type or occurs for the pair in regime (B). We say that has type , resp. type depending on the type of cancellation occurring in .

We want to use our knowledge of the *cancellation* on each to *modify* the *trivial* bound for : roughly speaking, we want to insert a factor in front of the terms of associated to or whenever . At this point, the localization tool in Lemma 8 will prove itself useful.

From the technical point of view, we implement the idea from the previous paragraph as follows. We consider a bump function localized on each : we impose on , outside and (for some universal constant ). Next, we construct on by putting

Note that for some constant depending only on the *fixed* objects , and . In particular, is *independent* of and the pair . The function was built up so that is a function taking values in such that and

has a definite contraction factor in front of the term associated to or whenever for some .

In summary, we manipulated some bump functions associated to to get a function such that:

- , ,
- for and given by the type of ,
- for or and not of the type of .

From the properties in the last two items, we see that the statement of Dolgopyat’s cancellation lemma 9 implies that

that is, is a bound for . By combining this fact with the property in the first item and the localization tool in Lemma 8, we have that is actually an *appropriate* bound for , i.e.,

Therefore, our task is reduced to prove that for some constant (independent of , and ). In this direction, a short (half-page) calculation (cf. page 193 in AGY paper) exploiting the cancellation mechanism for any shows that

and

Note that the estimate (3) goes in the *good* direction: it implies that

by the definition of the normalized transfer operator (and the fact that ), and we know that for sufficiently small.

On the other hand, the estimate (4) by itself is *not* sufficient to control in the desired way.

Fortunately, we know that covers “most” of the phase space and does not oscillate too much: more precisely, we have that is formed of intervals of size contained in some intervals covering the whole phase space (as it was said in the beginning of the proof) and .

By combining this facts with Gronwall’s inequality, it is possible to prove (after a short calculation, cf. pages 195 and 196 of AGY paper) that there exists a constant such that

Because the intervals are pairwise disjoint, the intervals cover , the density is bounded away from and , and is uniformly bounded (since the size of *both* and is ), the estimate above implies that there exists a constant such that

This new information (coming from the low oscillations of and the fact that covers almost all of ) permits to get the desired -contraction for . Indeed, this is a consequence of the following simple computation (resembling to “Peter-Paul inequality”). Denote by . By (3) and (4), we have

for any parameter . By applying (5) to the right-hand side of the previous inequality, we deduce that

for each . By taking so that , we see that the estimate above implies that

where .

Therefore, if we take small enough so that , then

This completes the proof of the lemma.

Evidently, the main point of Lemma 10 is that one can iterate it to obtain the contraction property in Proposition 2 in regime (B):

Corollary 11Let and . Suppose that is the first time such that does not exhibit high oscillation at scale , i.e.,Consider the integer fixed above. Let , and be the constants provided by Lemma 10, and denote by with , .

Then,

*Proof:* We set . By definition, where is the constant function . By successively applying Lemma 10, we obtain a sequence of pairs such that

By setting and by recalling that for all , we conclude from the previous estimate that

This proves the corollary.

At this point, the proof of Proposition 1 is complete. Indeed, this is so because the hypothesis of Proposition 2 is always satisfied in both regimes (A) (by Corollary 5) and (B) (by Corollary 11).

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Here, our main goal is to provide a little bit more of details on how this technique works by offering a “guided tour” through Sections 2 and 7 of a paper of Avila-Goüezel-Yoccoz.

For this sake, we organize this short series of posts as follows. In next section, we introduce a prototypical class of semiflows exhibiting exponential mixing. After that, we state the main exponential mixing result of this post (an analog of Theorem 7.3 of Avila-Goüezel-Yoccoz paper) for such semiflows, and we reduce the proof of this mixing property to a Dolgopyat-like estimate on weighted transfer operators. Finally, the next post of the series will be entirely dedicated to sketch the proof of the Dolgopyat-like estimate.

**1. Expanding semiflows **

Recall that a *suspension flow* is a semiflow , , associated to a *base dynamics* (discrete-time dynamical system) and a *roof function* in the following way. We consider where is the equivalence relation induced by , and we let be the semiflow on induced by

Geometrically, , flows up the point , , linearly (by translation) in the fiber until it hits the “roof” (the graph of ) at the point . At this moment, one is sent back (by the equivalence relation ) to the basis at the point , and the semiflow restarts again.

A more concise way of writing down is the following: denoting by , one defines where is the Birkhoff sum

and is the unique integer such that

In this post, we want to study the decay of correlations of *expanding semiflows*, that is, a suspension flow so that the base dynamics is an *uniformly expanding Markov map* and the roof function is a *good roof function with exponential tails* in the following sense.

Remark 1Avila-Gouëzel-Yoccoz work in greater generality than the setting of this post: in fact, they allow to be aJohn domainand they prove results forexcellent hyperbolic semiflows(which are more common in “nature”), but we will always take and we will study exclusively expanding semiflows (which are obtained from excellent hyperbolic semiflows by taking the “quotient along stable manifolds”) in order to simplify our exposition.

Definition 1Let , be the Lebesgue measure on , and be a finite or countable partition of modulo zero into open subintervals. We say that is anuniformly expanding Markov mapif

- is a
Markov partition: for each , the restriction of to is a -diffeomorphism between and ;- is
expanding: there exist a constant and, for each , a constant such that for each ;- has
bounded distortion: denoting by the inverse of the Jacobian of and by the set of inverse branches of , we require that is a function on each and there exists a constant such thatfor all and . (This condition is also called

Renyi conditionin the literature.)

Remark 2Araújo and Melbourne showed recently that, for the purposes of discussing exponential mixing properties (for excellent hyperbolic semiflows with one-dimensional unstable subbundles), the bounded distortion (Renyi condition) can be relaxed: indeed, it suffices to require that is a Hölder function such that the Hölder constant of is uniformly bounded for all .

Example 1Let be the finite partition (mod. ) of provided by the two subintervals , .The map given by for is an uniformly expanding Markov map (preserving the Lebesgue measure ).

An uniformly expanding map preserves an *unique* probability measure which is absolutely continuous with respect to the Lebesgue measure . Moreover, the density is a function whose values are bounded away from and , and is ergodic and mixing.

Indeed, the proof of these facts can be found in Aaronson’s book and it involves the study of the spectral properties of the so-called *transfer* (Ruelle-Perron-Frobenius) *operator*

(as it was discussed in Theorem 1 of Section 1 of this post in the case of a *finite* Markov partition , )

Definition 2Let be an uniformly expanding Markov map. A function is agood roof functionif

- there exists a constant such that for all ;
- there exists a constant such that for all and all inverse branch of ;
- is
nota –coboundary: it is not possible to write where is constant on each and is .

Remark 3Intuitively, the condition that is not a coboundary says that it is not possible to change variables to make the roof function into a piecewise constant function. Here, the main point is that we have to avoid suspension flows with piecewise constant roof functions (possibly after conjugation) in order to have a chance to obtain nice mixing properties (see this post for more comments).

Definition 3A good roof function hasexponential tailsif there exists such that .

The suspension flow associated to an uniformly expanding Markov map and a good roof function with exponential tails preserves the probability measure

on . Note that is absolutely continuous with respect to (because is absolutely continuous with respect to ).

Remark 4All integrals in this post are always taken with respect to or unless otherwise specified.

Remark 5In the sequel, AGY stands for Avila-Gouëzel-Yoccoz.

**2. Statement of the exponential mixing result **

Let be an expanding semiflow.

Theorem 4There exist constants , such that

for all and for all .

Remark 6By applying this theorem with in the place of , we obtain theclassicalexponential mixing statement:

Remark 7This theorem is exactly Theorem 7.3 in AGY paperexceptthat they work with observables and belonging to Banach spaces and which are slightly more general than (in the sense that ). In fact, AGY need to deal with these Banach spaces because they use their Theorem 7.3 to deduce a more general result of exponential mixing for excellent hyperbolic semiflows (see their paper for more explanations), but we will not discuss this point here.

The remainder of this post is dedicated to the proof of Theorem 4.

** 2.1. Reduction of Theorem 4 to Paley-Wiener theorem**

From now on, we fix two observables such that

Of course, there is no loss in generality here because we can always replace by if necessary.

In this setting, we want to show that the correlation function

For this sake, we will use the following classical theorem in Harmonic Analysis (stated as Theorem 7.23 in AGY paper):

Theorem 5 (Paley-Wiener)Suppose that can be analytically extended to a function defined on a strip in such a way thatLet be a bounded measurable function and denote by (defined for with ) the Laplace transform of .Then, there exists a constant and a full measure subset such that

for all .

In other words, the Paley-Wiener theorem says that a bounded measurable function decays exponentially whenever its Laplace transform admits a nice analytic extension to a vertical strip to the left of the imaginary axis .

In our context, we will produce such an analytic extension by writing the Laplace transform of the correlation function as an appropriate geometric series of terms depending on . In fact, the th term of this series will have a clear dynamical meaning: it will be related to the pieces of orbit hitting times the graph of the roof function.

More concretely, for each , we decompose the phase space as

where

- is the subset of points whose piece of orbit hits the graph of the roof function at least once,
- and is the subset of points whose piece of orbit does not hit the graph of the roof function.

Let us denote by

the corresponding decomposition of the correlation function (3).

The fact that the roof function has exponential tails implies that the probability of the event that the piece of orbit does not hit the graph of becomes exponentially small for large.

Thus, it is not surprising that the following calculation shows that decays exponentially fast as grows:

where (by the exponential tails condition on ).

In particular, the proof of Theorem 4 is reduced to show that decays exponentially. As we already mentioned, the basic idea to achieve this goal is to use the Paley-Wiener theorem and, for this reason, we want to write the Laplace transform as an appropriate series by decomposing the phase space accordingly to the number of times that a certain piece of orbit hits the graph of the roof function:

where is defined by (1). Note that this is valid for any with .

An economical way of writing down this series uses the *partial Laplace transform* of a function defined by the formula for . In this language, the previous identity gives that

By a change of variables (i.e., “duality”), we obtain that

where is the th iterate of the *weighted transfer operator*

Remark 8In general, we expect the terms of the series (4) to decay exponentially because is expanding and is “dual” to the (Koopman-von Neumann) operator defined by composing functions with .

This formula (stated as Lemma 7.17 in AGY paper) is the starting point for the application of Paley-Wiener theorem to . More precisely, we can exploit it to analytically extend into three steps:

- (a) Lemma 7.21 in AGY paper: can be extended to a neighborhood of any ;
- (b) Lemma 7.22 in AGY paper: can be extended to a neighborhood of the origin ;
- (c) Corollary 7.20 in AGY paper: can be extended to a strip for some small and large in such a way that the extension satisfies (for some constant depending on and ).

Geometrically, the first two steps permit to extend to *rectangles* of the form

for . Indeed, this is a consequence of (a), (b) and the compactness of the segment .

Of course, these steps are *not* sufficient to extend to a *whole* strip to the left of the imaginary axis (because there is *nothing* preventing as ), and this is why the third step is *crucial*.

In summary, the combination of the three steps (a), (b) and (c) (and the fact that the function is integrable) says that has an analytic extension to a strip satisfying the hypothesis of Paley-Wiener theorem 5.

Therefore, we have that decays exponentially (and, a fortiori, the correlation function also decays exponentially) *if* we can establish (a), (b) and (c).

** 2.2. Implementation of (a), (b) and part of (c) **

Observe that the series in (4) is bounded by

This hints that we should first compare the sizes of and to the sizes of and before trying to show the geometric convergence of this series (for certain values of ). In this direction, we have the following pointwise bound (compare with the equation (7.66) in AGY paper):

Lemma 6There exists a constant such that

for any function and for all with .

*Proof:* Recall that . Hence, the desired estimate is trivial for . The remaining case is dealt with by integration by parts. Indeed, we have

Since the right-hand side has boundary terms bounded by and an integral term bounded by

(as ), we see that the proof of the lemma is complete.

This pointwise bound permits to control (cf. Lemma 7.18 in AGY paper):

Corollary 7There exists a constant such that

for all with .

*Proof:* This is an immediate consequence of Lemma 6 and the fact that the function is integrable (by the exponential tails condition on ).

On the other hand, this pointwise bound is *not* adequate to control in terms of a geometric series. In fact, it is well-known that the weighted transfer operators *only* exhibit some contraction property when one *also* works with stronger norms than . In our current setting, it might be tempting to try to control the norm of in terms of the norm of . As we are going to see now, this does not quite work for directly (as the pointwise bound in Lemma 6 also involves the function which might be unbounded), but it does work for (compare with Lemma 7.18 of AGY paper):

Lemma 8There exists a constant such that

- ;
- ,

for all with .

*Proof:* Recall that .

It follows from Lemma 6 that

where denotes the constant function of value one. Since it is not hard to check that the operator acting on the space of functions is bounded for , the first item of the lemma is proved.

Let us now prove the second item of the lemma. For this sake, we write . In particular, the derivative has four terms: one can differentiate the term or or the limit of integration or (resp.). Let us denote by , , and (resp.) the terms obtained in this way.

The fourth term is bounded by

Similarly, the bounded distortion property for implies that

Finally, since (by definition of good roof function) and , we see that the third term is bounded by

and the first term is bounded by

(thanks to the estimate from the first item.)

This completes the proof of the lemma.

Remark 9This suggests that we should measure functions using the normAn important point in this lemma is that the norm of behavesdifferentlyfrom the norm of .

in order to get some uniform control on the operator : indeed, this norm allows to rewrite the previous lemma as

and this is exactly the statement of Lemma 7.18 in AGY paper.This norm will show up again in the statement of the Dolgopyat-like estimate.

Once we have Corollary 7 and Lemma 8, we are ready use the estimate (6) and some classical properties (namely, *Lasota-Yorke inequality* and *weak mixing for* ) in order to implement the step (a) of the “Paley-Wiener strategy”:

Lemma 9For any , there exists an open disk (of radius independent of and ) centered at such that has an analytic extension to .

*Proof:* It is well-known (cf. Lemma 7.8 in AGY paper) that the weighted transfer operator acting on satisfies a *Lasota-Yorke inequality*. We will come back to this point in the next post of this series, but for now let us just mention a key *spectral* consequence of a Lasota-Yorke inequality for .

By Hennion’s theorem (cf. Baladi’s book), a Lasota-Yorke inequality for implies that its essential spectral radius is and its *spectral radius* . In concrete terms, this means that there exists a constant such that the spectrum of is entirely contained in the ball *except* for possibly finitely many eigenvalues (counted with multiplicity) located in the annulus .

In other words, *if* one can show that has *no* eigenvalues of modulus , *then* the previous description of the spectrum of gives that for some constants and , and for all . Of course, since is a small analytic perturbation of for any in a small disk centered at , this implies that

for some constants , and for all , . In particular, by combining this estimate with Corollary 7, Lemma 8 and (6), we obtain that in this setting the series

defines an analytic extension to of .

In summary, we have reduced the proof of the lemma to the verification of the fact that has no eigenvalues of modulus (when ). As it turns out, this is an easy consequence of the spectral characterization of the weak-mixing property for the expanding semiflow : indeed, this property says that the Koopman-von Neumann operator given by composition with has no eigenvalues of modulus , and it is not difficult to see that this means that has no eigenvalues of modulus .

This proves the lemma.

Remark 10A more direct proof of this lemma (without relying on the weak-mixing property for ) can be found in Lemma 7.21 of AGY paper.

Next, let us adapt the argument above to perform the step (b) of the Paley-Wiener strategy:

Lemma 10There exists an open disk (of radius independent of and ) centered at such that has an analytic extension to .

*Proof:* The transfer operator has a simple eigenvalue (cf. Aaronson’s book). In particular, the argument used to prove the previous lemma does not work (i.e., it is simply *false* that decays exponentially as ).

Nevertheless, we can overcome this difficulty as follows. For in a small open disk centered at , is an analytic perturbation of . Thus, has an eigenvalue close to , and we can write

where is the spectral projection to the eigenspace generated by the normalized eigenfunction (with ) associated to , and . Furthermore, the spectral properties of mentionned during the proof of Lemma 9 also tell us that there exist uniform constants and such that for all and .

In other terms, after we remove from the component associated to the eigenvalue , we obtain an operator with nice contraction properties.

At this point, the basic idea is to “repeat” the argument of the proof of Lemma 9 with replaced by . In this direction, we rewrite the series (4) as

(Here, we used that , , and )

Observe that the series converges for all thanks to Corollary 7, Lemma 8 and the fact that with for all .

It follows that we can use the previous equation to define an analytic extension of to *if* we can control the term . In other words, the proof of the lemma is reduced to show that

Note that this is a completely obvious task for because , i.e., the analytic function

has a *pole* at .

Fortunately, the order of pole at of this function can be shown to be one by the following calculation. Since is an analytic perturbation of , we have that and . In particular,

where is the absolutely continuous invariant probability measure of . This means that has derivative at , and, *a fortiori*, the pole of at has order one.

Thus, the function (8) is analytic on . Moreover, it also follows that the function (8) can be analytically extended to if we show that

has a zero at . This last fact is not hard to check: by definition, is a constant multiple of the function and , so that

and our assumption (2) was precisely that .

This proves the lemma.

Closing this post, let us reduce the step (c) to the following *Dolgopyat-like estimate* (compare with Proposition 7.7 in AGY paper):

Proposition 11There exist , , and such that

for all with , , . (Here, is the norm introduced in Remark 9.)

The proof of this proposition will occupy the next post of this series. For now, let us implement the step (c) of the Paley-Wiener strategy assuming Proposition 11.

We want to use the formula (4) to define a suitable analytic extension

of to a strip of the form .

By (6), Proposition 11, Corollary 7 and Lemma 8, we have

for all with and .

This proves that is an analytic extension of to such that , which are exactly the properties required in the step (c).

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Lemma 1Let and be two constants such that and for all .Let be a diffeomorphism from the closed unit ball of into its image.Then, for each , the -dimensional Hausdorff measure at scale of satisfies

Remark 1In fact, this is not the version of the lemma used in practice by Palis, Yoccoz and myself. Indeed, for our purposes, we need the estimate

where is the ball of radius centered at the origin and is a diffeomorphism such that and for . Of course, this estimate is deduced from the lemma above by scaling, i.e., by applying the lemma to where is the scaling .

Nevertheless, we are not completely sure if we should write down an article just with our current partial results on non-uniformly hyperbolic horseshoes because our feeling is that these results can be significantly improved by the following heuristic reason.

In a certain sense, Lemma 1 says that one of the “worst” cases (where the estimate (1) becomes “sharp” [modulo the multiplicative factor ]) happens when is an affine hyperbolic conservative map (say ): indeed, since , the most “economical” way to cover using a countable collection of sets of diameters is basically to use squares of sizes (which gives an estimate ).

However, in the context of (expectional subsets of stable sets of) non-uniformly hyperbolic horseshoes, we deal with maps obtained by successive compositions of affine-like hyperbolic maps *and* a certain *folding* map (corresponding to “almost tangency” situations). In particular, we work with maps which are *very different* from affine hyperbolic maps and, thus, one can expect to get slightly *better* estimates than Lemma 1 in this setting.

In summary, Jacob, Jean-Christophe and I hope to improve the results announced in this survey here, so that Lemma 1 above will become a “deleted scene” of our forthcoming paper.

On the other hand, this lemma might be useful for other purposes and, for this reason, I will record its (short) proof in this post.

**1. Proof of Lemma 1 **

The proof of (1) is based on the following idea. By studying the intersection of with dyadic squares on , we can interpret the measure as a sort of -norm of a certain function. Since , we can control this -norm in terms of the and norms (by interpolation). As it turns out, the -norm, resp. -norm, is controlled by the features of the derivative , resp. Jacobian determinant , and this morally explains the estimate (1).

Let us now turn to the details of this argument. Denote by and its boundary. For each integer , let be the collection of *dyadic squares* of level , i.e., is the collection of squares of sizes with corners on the lattice .

Consider the following recursively defined cover of . First, let be the subset of squares such that

Next, for each , we define inductively as the subset of squares such that is *not* contained in some for , and intersects a *significant portion* of in the sense that

In other words, we start with and we look at the collection of dyadic squares of level intersecting it in a significant portion. If the squares in suffice to cover , we stop the process. Otherwise, we consider the dyadic squares of level not belonging to , we divide each of them into four dyadic squares of level , and we build a collection of such dyadic squares of level intersecting in a significant way the remaining part of not covered by , etc.

Remark 2In this construction, we are implicitly assuming that is not entirely contained in a dyadic square . In fact, if , then the trivial bound (for ) is enough to complete the proof of the lemma.

In this way, we obtain a countable collection covering such that and

By thinking of this expression as a -norm and by applying interpolation between the and norms, we obtain that

This reduces our task to estimate these and norms. We begin by observing that the -norm is easily controlled in terms of the Jacobian of (thanks to the condition (2)):

for any . In particular, we have that

From this estimate, we see that the -norm satisfies

Thus, we have just to estimate the series . We affirm that this series is controlled by the derivative of . In order to prove this, we need the following claim:

**Proof of Claim.** Note that can not contain : indeed, since for some dyadic square of level (and, thus, ), if , then , a contradiction with the definition of . Because we are assuming that is not contained in (cf. Remark 2) and we also have that intersects (a significant portion of) , we get that

For the sake of contradiction, suppose that . Since intersects , the -neighborhood of contains . This means that

- (a) either is contained in
- (b) or is disjoint from

However, we obtain a contradiction in both cases. Indeed, in case (a), we get that a dyadic square of level containing satsifies

a contradiction with the definition of . Similarly, in case (b), we obtain that

a contradiction with (2).

This completes the proof of the claim.

Coming back to the calculation of the series , we observe that the estimate (7) from the claim and the fact that imply:

By plugging this estimate into (6), we deduce that the -norm verifies

Finally, from (3), (4), (5) and (8), we conclude that

This ends the proof of the lemma.

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These lectures by Alex Wright made Eduard Looijenga ask if some “remarkable” translation surfaces could help in solving the following question.

Let be a ramified finite cover of the two-torus (say branched at only one point ). Denote by the subspace of generated by the homology classes of all simple closed loops on covering such a curve on .

**Question 1.** Is it true that one *always* has in this setting?

By following Alex Wright’s advice, Eduard Looijenga wrote me asking if I knew the answer to this question. I replied to him that my old friend *Eierlegende Wollmilchsau* provided a *negative* answer to his question, and I directed him to the papers of Forni (from 2006), Herrlich-Schmithüsen (from 2008) and our joint paper with Yoccoz (from 2010) for detailed explanations.

In a subsequent email, Eduard told me that my answer was a good indication that notable translation surfaces could be interesting for his purposes: indeed, the Eierlegende Wollmilchsau is *precisely* the example described in the appendix of a paper by Andrew Putman and Ben Wieland from 2013 where Question 1 was originally solved.

After more exchanges of emails, I learned from Eduard that his question was motivated by the attempts of Putman-Wieland (in the paper quoted above) to attack the following conjecture of Nikolai Ivanov (circa 1991):

**Conjecture (Ivanov).** Let and . Consider a finite-index subgroup of the mapping-class group of isotopy classes of homeomorphisms of a genus surface fixing pointwise a set of marked points. Then, there is no surjective homomorphism from to .

Remark 1This conjecture came from the belief that mapping-class groups should behave in many aspects like lattices in higher-rank Lie groups and it is known that such lattices do not surject on because they satisfy Kazhdan property (T). Nevertheless, Jorgen Andersen recently proved that the mapping-class groups do not have Kazhdan property (T) when .

Remark 2It was proved by John McCarthy and Feraydoun Taherkhani that the analog for of Ivanov’s conjecture fails.

In fact, Putman-Wieland proposed the following strategy to study Ivanov’s conjecture. First, they introduced the following conjecture:

**Conjecture (Putman-Wieland).** Fix and . Given a finite-index characteristic subgroup of the fundamental group of a surface of genus with punctures , denote by the associated finite cover, and let be the compact surface obtained from by filling its punctures.

Then, the natural action on of the group of lifts to of isotopy classes of diffeomorphisms of fixing pointwise has *no* finite orbits.

Remark 3This conjecture is closely related (for reasons that we will not explain in this post) to a natural generalization of Question 1 to general ramified finite covers .

Remark 4The analog of Putman-Wieland conjecture in genus is false: the same counterexample to Question 1 (namely, the Eierlegende Wollmilchsau) serves to answer negatively this genus 1 version of Putman-Wieland conjecture.

Remark 5In the context of Putman-Wieland conjecture, one has a representation (induced by the lifts of elements of to ). This representation is called ahigher Prym representationby Putman-Wieland. In this language, Putman-Wieland conjecture asserts that higher Prym representations have no non-trivial finite orbits when and .

Secondly, they proved that:

Theorem 1 (Putman-Wieland)Fix and .

- (a) If Putman-Wieland conjecture holds for every finite-index characteristic subgroup of , then Ivanov conjecture is true for any finite-index sugroup of .
- (b) If Ivanov conjecture holds for every finite-index subgroup of , then Putman-Wieland conjecture is true for any finite-index characteristic subgroup of .

Moreover, if Ivanov conjecture is true for all finite-index subgroups of for all , then it is also true for all finite-index subgroups of with , .

In other words, Putman-Wieland proposed to approach an algebraic problem (Ivanov conjecture) via the study of a geometric problem (Putman-Wieland conjecture) because these two problems are “essentially” equivalent.

In particular, this gives the following concrete route to establish Ivanov conjecture:

- (I) if we want to show that Ivanov conjecture is
*true*for all and , then it suffices to prove Putman-Wieland conjecture for (and all ); indeed, this is so because item (a) of Putman-Wieland theorem would imply that Ivanov conjecture is true for (and all ) in this setting, and, hence, the last paragraph of Putman-Wieland theorem would allow to conclude the validity of Ivanov conjecture in general. - (II) if we want to show that Ivanov conjecture is
*false*for some and , then it suffices to construct a counterexample to Putman-Wieland conjecture for and .

Once we got at this point in our email conversations, Eduard told me that Question 1 was just a warmup towards his main question:

**Question 2. **Are there remarkable translation surfaces giving counterexamples to Putman-Wieland conjecture?

By inspecting my list of “preferred” translation surfaces, I noticed that I knew such an example: in fact, there is exactly one member in a family of translation surfaces that I’m studying with Artur Avila and Jean-Christophe Yoccoz (for other purposes) which is a counterexample to Putman-Wieland conjecture in genus (and ).

In other words, one of the translation surfaces in a forthcoming paper joint with Artur and Jean-Christophe answers Question 2.

Remark 6This shows that Putman-Wieland’s strategy (I) above doesnotwork (because their conjecture is false in genus ). Of course, this doesnotmean that Ivanov conjecture is false: in fact, by Putman-Wieland strategy (II), one needs a counterexample to Putman-Wieland conjecture in genus (rather than in genus ). Here, it is worth to point out that Artur, Jean-Christophe and I havenogood candidates of counterexamples to Putman-Wieland conjecture in genus and/or Ivanov conjecture.

Below the fold, we focus on the case and of Putman-Wieland conjecture.

**1. A genus cover of a genus surface **

Let be the genus surface associated to the Riemann surface . The genus surface corresponding to the Riemann surface has the structure of a triple cover given by . Observe that is unramified off the six (Weierstrass) points of located at the five roots of unit , , and the point at infinity .

Recall that the ramified finite cover corresponds to a finite-index subgroup of where is a genus surface with punctures and is a point of located at .

It is possible to check that is *not* a characteristic subgroup of . Nevertheless, we can easily construct a subgroup of such that is a finite-index characteristic subgroup of . Indeed,

is a subgroup of which is characteristic in . Furthermore, has finite-index in because has the same (finite) index of for all and has only finitely many subgroups of a given index (since is finitely generated).

Denote by the compact surface associated to the finite ramified cover of induced by , and let be the mapping-class group of . Since is characteristic, we can lift any element of to a mapping-class of , so that we have a higher Prym representation .

Theorem 2There exists a eight-dimensional subspace such that the orbit of any under the higher Prym action of on is finite. In particular, the Putman-Wieland conjecture is false in the case and .

We will deduce this theorem as a consequence of the following result:

Theorem 3There exist a eight-dimensional subspace and a finite-index subgroup of with the following properties. Any element of lifts to a mapping-class of and the orbit of any under the corresponding representation is finite.

The proof of Theorem 3 relies on the unusual features of the Hodge filtration of .

**2. Proof of Theorem 2 assuming Theorem 3**

Since is a subgroup of , we have that the cover associated to *factors* through the cover (associated to ), that is, we have a cover such that the composition is the cover corresponding to .

Given a eight-dimensional subspace and a finite-index subgroup of as in the statement of Theorem 3, let be the subspace of cohomology cycles projecting to which are also invariant under the whole group of deck transformations of .

By Theorem 3, the natural action of on factors a finite group of matrices. By construction, all orbits of the action of on are finite. Since is a finite-index subgroup of , it follows that all orbits of the action of on are also finite.

Finally, since the homology group is in a natural duality with the cohomology group , our reduction of Theorem 2 to Theorem 3 is complete.

**3. Proof of Theorem 3**

Let be the automorphism where generating the group of deck transformations of the cover where is the Riemann sphere and .

Note that factors : indeed, where is the natural projection from to the quotient of by its hyperelliptic involution , .

Since has genus , the elements of commute with the hyperelliptic involution : this is a very special property of the genus setting whose proof follows from the results in this paper here (see also page 77 of Farb-Margalit book [while paying attention that our convention differs from them because our mapping-class groups are required to fix pointwise each puncture]).

It follows that the elements with form a finite-index subgroup of such that the lift to of any such commutes with the automorphism (in fact, this is so because projects under to the hyperelliptic involution of ).

By construction, acts on and our task is to show the existence of a eight-dimensional subspace of such that the -orbit of any is finite.

For this sake, we start by analyzing the action of on . Here, the crucial point is that was built in such a way that all of its elements commute with . In particular, the action of preserves each summand of the decomposition

into the eigenspaces associated to the eigenvalues of . (Note that the eigenspace is trivial because ).

Recall that the action of on preserves the intersection form . Since each eigenspace has a Hodge decomposition

and the intersection form is positive definite on the space of holomorphic -forms and negative definite on the space of anti-holomorphic -forms, we have that acts on via a indefinite unitary group of a pseudo-Hermitian form of signature where

In our context, is associated to the curve , so that

is an explicit basis of the space of holomorphic -forms on . From this, we infer that and (and, in general, for each ).

In other words, , , and acts on the eight-dimensional complex subspace

via (a subgroup of) the *compact* group .

Next, we study the action of on . We begin by noticing that the eight-dimensional complex subspace is defined over . In fact, this is a consequence of the following elementary observation (from Galois theory): is the sum of *all* eigenspaces associated to *all* primitive th roots of unity .

Since is defined over , it intersects into a lattice of rank . In particular, acts on via (a subgroup of) the symplectic group because respects the symplectic intersection form on .

In summary, we proved that:

- on one hand, acts on via the compact group ;
- on the other hand, acts on via .

In other terms, acts on through a *compact* subgroup of the *discrete* group , i.e., acts on the eight-dimensional subspace through a *finite* subgroup of symplectic matrices.

It follows that the -orbit of any is finite, so that the proof of Theorem 3 is complete.

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As it is always the case with Sébastien’s expositions, he managed to communicate very clearly the ideas of a mathematically profound subject (and, by the way, this topic is not directly related to his excellent Bourbaki seminar talk from March 21st).

In the sequel, I’ll transcript my lecture notes for Sébastien’s talk. Of course, all errors and mistakes are my entire responsibility.

**1. Introduction **

Let us warmup by giving a proof of the following theorem:

Theorem 1 (Kohlberg-Neyman (1981))Let be a weak contraction of the Euclidean space in the sense that

Then, the sequencefor all .

converges as .

Remark 1The origin can be replaced by any point because

so that

As the reader might suspect, the fact that such an “innocent-looking” result was proved only in 1981 (in this paper here) indicates that its proof is not easy to find if we don’t use the “correct” setup.

For the purposes of this post, we will show Theorem 1 using a argument of Karlsson (from 2001).

*Proof:* The argument has two steps:

- the first step is to show that the distance of to the origin converges;
- the second step is to control the direction of .

The convergence of the distance

follows from the *subadditivity* of .

More precisely, since is a weak contraction, we have that

From this, it is not hard to see that converges to .

Indeed, given , we fix such that

Next, given , we write with . From the subadditivity of , we have that

In particular,

for all sufficiently large (i.e., ). In other words, we have that

for any , that is, , as desired.

Next, let us control the direction of .

Observe that the case when is easy: by definition, the sequence converges to the origin , so that the proof of the theorem is complete in this situation.

Thus, it remains only to consider the case when , i.e., the sequence goes to with positive linear speed.

Fix a sequence such that as . Note that, by definition, for each , one has

as . This means that we can consider a sequence of “records” of the excursion of towards , i.e.,

for all .

Denote by a linear form on of norm so that . Because has norm and is a weak contraction, we see that

for all .

Since is a sequence of records, we conclude that

for all .

Now, we note that, up to taking a subsequence, one can assume that the sequence of linear forms of norm converges (in the weak- topology) to a linear form .

By construction, for all . Geometrically, this means that the sequence stays to the right of a sequence of parallel hyperplanes moving with linear speed to infinity.

Indeed, this is easily visualized in two dimensions: after rotating the kernel of in , we can suppose that . In this case, the inequality means that belongs to the half-plane .

From this geometric input, it is not hard to complete the proof of the theorem.

In fact, we have that, on one hand, for all , and, on the other hand, for each , we have that for all sufficiently large (because ).

From the strict convexity of the Euclidean ball , we obtain that belongs to the small lenticular regions whose geometries forces the direction of to be -close to the unit vector perpendicular to the kernel of . Since is arbitrary, we conclude that the directions of converge (to in our current situation).

Of course, this convergence together with the fact that as finishes the proof of the theorem.

Remark 2Note that, except for the last part of the proof (where the geometry [strict convexity] of balls entered in the discussion), Karlsson’s argument can be generalized to abstract Banach spaces in the place of : for instance, the existence of the linear functionals follows from Hahn-Banach theorem, and the weak- convergence of a subsequence of is a consequence of Banach-Alaoglu theorem.

Remark 3The statement of Theorem 1 is very sensitive on the choice of the norm. For example, this theorem is false for equipped with the supremum norm . In fact:

- Karlsson’s argument fails because the information
does not impose strong constraints in the direction of since the “ball”

is now a square, and the “lenticular region”

is a now rectangle of width and height .

- A weak -contraction such that does not converge can be constructed as follows. Let and . Consider a sequence of times such that . Using this sequence of times, we define so that follows a straight line segment in the direction for time , then a straight line segment in the direction for time , then a straight line segment in the direction for time , etc. (e.g., for , for , etc.); Since consists of straight line segments of slopes or , one can see that is a geodesic ray for the supremum norm (such that ); From , we define a weak -contraction by letting
where for . In this context, and, for an appropriate choice of times , one can check that does not converge because its direction keeps oscillating between and .

**2. Horofunctions **

After this quick review of Karlsson’s proof of Kohlberg-Neyman theorem for weak contractions in Euclidean spaces, let us pass now to the study of weak contractions in metric spaces.

For this sake, we need the following tool (playing the role of “linear functionals”):

Definition 2Let be a metric space and . We say that a function is a horofunction (or Busemann function) if there exists a sequence such that

Remark 4By definition, . Furthermore, is a -Lipschitz function (because it is the limit of the -Lipschitz functions ).

Example 1Consider the Euclidean space and let , . The level sets of the function are the Euclidean balls centered at : more precisely, takes the value on the circle of radius centered at . From this, one can see that converges to the linear functional .In this way, we see that linear functionals in Euclidean spaces are particular cases of horofunctions.

Example 2Consider the plane equipped with the supremum norm . Let and . A direct inspection reveals that the level sets of are translations along the diagonal of the first quadrant of , i.e., where . In particular, converges to the horofunction .

Remark 5It is possible to show that any horofunction on a normed vector space can be estimated from below by a linear functional. In particular, the level sets of horofunctions provide more information on the location of points than the level sets of linear functionals.

Example 3In the case of the hyperbolic plane, the horofunctions are the classical Buseman functions (whose level sets are the horospheres).

The same argument used by Karlsson to prove Theorem 1 allowed him to show the following result:

Theorem 3 (Karlsson (2002))Then,Let be a separable metric space, a weak contraction, and .

and there exists a horofunction such that for all .

**3. Iterated systems of contractions **

Note that Karlsson’s theorem gives a geometrical description of the orbits of a *single* weak contraction, but one might wonder about the behavior of “random” compositions (*cocycle*) of several weak contractions.

In this direction, Gouëzel and Karlsson proved the following theorem:

Theorem 4 (Gouëzel-Karlsson)Denote by . Then, there exists such that, for -almost every , all , and some horofunction , we have thatLet be a ergodic transformation, be a metric space and an (integrable) cocycle (where is the set of weak contractions of ).

and

as .

Remark 6There is no hope to get this kind of convergence result if we compose the weak contractions in the other way around in the definition of . In fact, this is not hard to see in the context of the composition of random (large) hyperbolic matrices in acting on the hyperbolic disk : since a large hyperbolic matrix tends to concentrate a big chunk of near a boundary point associated to the unstable direction of , we have that the compositions

of random large hyperbolic matrices will most likely take the origin of near the random boundary point , and, thus will not converge; on the other hand, the compositions

of random large hyperbolic matrices will most likely take the random point near the deterministic boundary point and this is why we expect convergence of .

Before saying a few words about the proof of Gouëzel-Karlsson theorem, let us put it into historical perspective by citing a couple of previous related results:

- Karlsson and Ledrappier showed in 2006 the validity of Gouëzel-Karlsson theorem in the special case of a cocycle of
*isometries*(by exploiting in particular the fact that the composition of a horofunction and a isometry is still a horofunction); - Karlsson and Margulis proved in 1999 the following slightly weaker version of Gouëzel-Karlsson theorem: under the same assumptions of Theorem 4, for each , there exists an horofunction such that
converges to a value in the interval as .

Let us now conclude this post with a sketch of proof of Theorem 4.

The first observation is that the sequence

is a *subadditive cocycle* in the sense that

Here, we used that takes its values in the set of weak contractions and was defined as (in this order).

In this setting, we can apply Kingman’s subadditive ergodic theorem to deduce the convergence of :

Theorem 5 (Kingman (1967))Let be a (integrable) subadditive cocycle. Then,

almost surely and in . (Here, is independent of .)

Therefore, the proof of Gouëzel-Karlsson theorem will be complete once we construct a horofunction with the properties in the statement of Theorem 4.

Unfortunately, the information provided by Kingman’s theorem is *not* sufficient to build up the desired . For this reason, Gouëzel and Karlsson were “forced” to show the following *improvement* of Kingman’s subadditive ergodic theorem:

Theorem 6 (Gouëzel-Karlsson)Let be a (integrable) subadditive cocycle such that

Then, for -almost every , there are as and as such that, for all and all ,

An important point of this theorem is that is *independent* from ! In particular, this fact can be exploited to build up a horofunction (obtained as the limit of a subsequence of suitable functions defined in the spirit of Karlsson’s proof of Kohlberg-Neyman theorem) such that

for all . Of course, since as , the previous estimate permits to conclude the proof of Gouëzel-Karlsson theorem.

Finally, the proof of Gouëzel-Karlsson improvement of Kingman’s theorem is similar in spirit to the usual proofs of Kingman’s result (based on the combinatorics of pieces of orbits of where the values of the cocycle fluctuates near a given value).

However, the technical details are somewhat *intrincate* and, for this reason, Sébastien decided that it was a better idea to explain why the *naive* approach based on the study of times where the cocycle attains its “records” (as in Karlsson’s proof of Kohlberg-Neyman theorem) does *not* work.

So, we close this post by following him in the explanation of two naive strategies that *fail* in proving Theorem 6.

We begin by taking a sequence as . By definition, for -almost every , we have that

Thus, it makes sense to consider a sequence of successive “records”, i.e.,

for all . Of course, the previous inequality can be rewritten as

which *looks like* the conclusion of Theorem 6 *except* that the argument of is instead of !

At this point, we have the impression to be “close” to show Theorem 6, but this is not the case. Indeed, suppose that we try to overcome the difficulty of the previous paragraph by making the “change of variables” (in hope of finding in the place of in the argument of ). It is not hard to check that is a subadditive cocycle with respect to . In particular, we can repeat our discussion with replaced by in order to find a sequence of “records”/good times for almost every .

By doing so, we have that almost every belongs to infinitely many of the subsets

Logically, we are not directly interested in the cocycle , but rather on . In other words, even though we got some information about the sets , we are really interested in the subsets

Here, it is tempting to *conjecture* that the fact that almost every belongs to infinitely many (from which case we would be able to deduce Theorem 6) is an immediate consequence of the fact that almost every belongs to infinitely many . Unfortunately, this conjecture is simply *wrong* if we do not have extra information about the structure of the subsets : for example, let be a fixed subset of *positive but not full* -measure; the ergodicity of implies that -almost every belongs to infinitely many , but it is obvious that the complement of is a subset of positive -measure subset consisting of which do not belong to a single !

In summary, a naive combination of the “records” strategy and a change of variables do not lead us anywhere close to showing Theorem 6, and, in fact, a more sophisticated combinatorial strategy is needed here.

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In fact, our goal today is to revisit Luigi Ambrosio’s Bourbaki seminar talk entitled “*The regularity theory of area-minimizing integral currents (after Almgren-DeLellis-Spadaro)*”. Here, besides the original works of Almgren and DeLellis-Spadaro (in these five papers here), the main references are the video of Ambrosio’s talk and his lecture notes (both in English).

**Disclaimer.** As usual, all errors and mistakes are my entire responsibility.

**1. Introduction **

This post is centered around solutions to the so-called Plateau’s problem.

A formulation of Plateau’s problem in *dimension* and *codimension* is the following. Given a -dimensional Riemannian manifold and a -dimensional compact embedded oriented submanifold (without boundary), find a -dimensional embedded oriented submanifold with boundary such that

for all oriented -dimensional submanifold with . (Here, denotes the -dimensional volume of ).

This formulation of Plateau’s problem allows for *several* variants. Moreover, the solution to Plateau’s problem is *very sensitive* on the precise mathematical formulation of the problem (and, in particular, on the dimension and codimension ).

Example 1The works of Douglas and Radó (based on the conformal parametrization method and some compactness arguments) provided a solution to (a formulation of) Plateau’s problem when and the boundary is circular (i.e., is parametrized by the round circle ). Unfortunately, the techniques employed by Douglas and Radó do not work for arbitrary dimension and codimension .

Example 2Let us consider the case , defined asThe following example gives an idea on the difficulties that one might found while trying to solve Plateau’s problem (in the formulation given above).The

singularimmersed disk

satisfies and the so-called calibration method can be applied to prove that

for all smooth oriented -dimensional submanifold with . (Here, stands for the -dimensional Hausdorff measure on .)

The example above motivates the introduction of *weak solutions* (including immersed submanifolds) to Plateau’s problem, so that the main question becomes the *existence* and *regularity* of weak solutions.

This point of view was adopted by several authors: for example, De Giorgi studied the notion of *sets of finite perimeter* when , and Federer and Fleming introduced the notion of *currents*.

Remark 1The “PDE counterpart” of this point of view is the study of existence and regularity of weak solutions of PDEs in Sobolev spaces.

In arbitrary dimensions and codimensions, De Giorgi’s regularity theory provides weak solutions to Plateau’s problem which are smooth on *open and dense* subsets.

As it was pointed out by Federer, this is *not* a satisfactory regularity statement. Indeed, De Giorgi’s regularity theory allows (in principle) weak solutions to Plateau’s problem whose *singular set* (i.e., the subset of points where the weak solution is not smooth) could be “large” in the sense that its (-dimensional) Hausdorff measure could be positive.

In this context, Almgren wrote two preprints which together had more than 1000 pages (published in posthumous way here) where it was shown that the singular set of weak solutions to Plateau’s problem has codimension at least:

Theorem 1 (Almgren)If is area-minimizing (i.e., a weak solution to Plateau’s problem), then there exists a closed subset of such that:Let be an integral rectifiable -dimensional current in a -dimensional Riemannian manifold .

- has codimension : the Hausdorff dimension of is , and
- is the singular set of : the subset is induced by a smooth oriented -dimensional submanifold of .

We will explain the notion of area-minimizing integral rectifiable currents (appearing in the statement above) in a moment. For now, let us just make some historical remarks. Ambrosio has the impression that some parts of Almgren’s work were not completely reviewed, even though several experts have used some of the ideas and techniques introduced by Almgren. For this reason, the simplifications (of about ) and, more importantly, improvements of Almgren’s work obtained by DeLellis-Spadaro in a series of five articles (quoted in the very beginning of the pots) were very much appreciated by the experts of this field.

Closing this introduction, let us present the plan for the remaining sections of this post:

- the next section reviews the construction of solutions to the generalized Plateau problem via Federer-Fleming compactness theorem for integral rectifiable currents;
- then, in the subsequent section, we revisit some aspects of the regularity theory of weak solutions to Plateau’s problem in codimension ; in particular, we will see in this setting a
*stronger*version of Theorem 1; - after that, in the last section of this post, we make some comments on the works of Almgren and DeLellis-Spadaro on the regularity theory for Plateau’s problem in codimension ; in particular, we will sketch the proof of Theorem 1 above.

Remark 2For the sake of exposition, from now on we will restrict ourselves to the study of Plateau’s problem in the case of Euclidean ambient spaces, i.e., . In fact, this is not a great loss in generality because the statements and proofs in the Euclidean setting can be adapted to arbitrary Riemannian manifolds with almost no extra effort.

**2. Federer-Fleming theory of currents **

Definition 2An integral rectifiable -dimensional current in is a triple where:

- is a countably -rectifiable set, i.e., where has zero -dimensional Hausdorff measure () and for all one has that for some -dimensional oriented submanifold ;
- is an orientation of : is a measurable map such that for each and for -almost every , one has that
for an oriented orthonormal basis . In other terms, is the approximate tangent space of at ;

- is the multiplicity: is a measurable -integrable function such that intuitively describes how many copies of oriented pieces of the current has near a -typical point (for each ).

An integral rectifiable -dimensional current induces a continuous linear functional on the space of compactly supported and smooth differential -forms on via the formula:

Remark 3In general, a -dimensional current is an element of the dual of (i.e., a continuous linear functional on ).

Example 3The notion of integral rectifiable -dimensional current generalizes the definition of an oriented smooth compact -dimensional submanifold . Indeed, given such a , we have that the triple is an integral rectifiable -dimensional current (where is the orientation of and is the constant function.)

The notions of *boundary*, *mass* and *support* of an integral rectifiable -dimensional current are defined as follows.

Definition 3The boundary is the -dimensional current satisfying Stokes formula:

The mass of is .where is the exterior derivative of .

The support of is the support of the measure , i.e., .

Example 4For the current associated to an oriented smooth compact manifold with boundary , the boundary is

the mass is , and the support is .

These definitions motivate the following generalization of Plateau’s problem (formulated in the introduction of this post):

*Generalized Plateau’s problem*. Let be an integral rectifiable -dimensional current compactly supported in such that . Find an integral rectifiable -dimensional current such that and

for all integral rectifiable -dimensional currents with .

Of course, the main point in generalizing Plateau’s problem is that one can *always* solve the generalized Plateau’s problem!

More precisely, a classical “cone construction” shows that there are always integral rectifiable currents with . In particular, it makes sense to consider

Next, the space of currents has a weak- topology associated to the evaluation maps

for each . As it is expected from weak- topologies (cf. Banach-Alaoglu theorem), we have a compactness property allowing us to extract a convergent subsequence from any area-minimizing sequence in the sense that are integral rectifiable currents with and . In particular, .

Furthermore, it is possible to prove that the mass depends on a lower semicontinuous way on the current. Since and , this means that satisfies

This *almost* puts us in position to solve the generalized Plateau problem. Indeed, the weak- limit constructed above solves the generalized Plateau problem *except* for the fact that it is not obvious at all that the current is *integral rectifiable*!

Fortunately, it turns out that is an integral rectifiable current: this is a consequence of the following compactness theorem for integral rectifiable currents of Federer-Fleming.

Theorem 4 (Federer-Fleming)Let be a sequence of integral rectifiable -dimensional currents converging to . Suppose that

Then, is also an integral rectifiable -dimensional current.

In summary, Federer-Fleming’s theorem is a key result permitting us to find solutions to the generalized Plateau’s problem.

Definition 5A solution to the generalized Plateau problem is called an area-minimizing integral rectifiable current.

Once the existence of area-minimizing currents is established, one can hope to answer the original Plateau problem by studying their *regularity*.

More concretely, let

be the set of (*interior*) regular points of , and let

be the *singular* set of . In this language, one wants to understand the *size* of .

Remark 4By requiring to be a -dimensional submanifold in the definition of , we are implicitly skipping the interesting question ofboundary regularityof ! In fact, the study of the boundary regularity of area-minimizing currents is a delicate problem (which is very sensible to the regularity of in the statement of the generalized Plateau problem): one disposes of results in the codimension case, but there is no satisfactory boundary regularity theory in arbitrary codimensions.

We divide the analysis of the size of into two parts. In the next section, we will review De Giorgi’s regularity theory in codimension : in particular, we will see that has codimension for any area-minimizing integral rectifiable current . Then, we dedicate the final section to present a rough sketch of the theorems of Almgren and DeLellis-Spadaro.

**3. Plateau’s problem in codimension **

In this section, we follow De Giorgi’s original approach by studying the regularity of sets with finite perimeter. In terms of currents, if we consider the -dimensional current

associated to via integration of -differential forms and we require boundary current has finite mass, then it is possible to prove that has the form

Remark 5The fact that the integral rectifiable current has multiplicity is a very peculiar feature of the codimension case!

In this setting, is called the *essential boundary* of and the *perimeter* of is .

For later use, we denote by the *approximate unit normal* to , i.e., for -a.e. , we take to be the unique normal vector to the -plane associated to which is positively oriented (i.e., where stands for the canonical basis of ).

** 3.1. De Giorgi’s almost everywhere regularity theorem **

In order to study the size of the singular set of a locally perimeter minimizing set , De Giorgi observed that the *deviations* (actually, variance) of the approximate unit normal from its mean are a fundamental tool.

More concretely, De Giorgi introduced the following quantity called the *excess*:

where

By definition, on , the excess is the variance of from its mean .

In this context, De Giorgi showed that a small excess permits to detect regular points of locally perimeter minimizing sets:

Theorem 6 (De Giorgi)Let be a set locally minimizing perimeter in an open set (i.e., for every such that the symmetric difference is compactly contained in a ball with closure ).For each , there exists a dimensional threshold with the following property.Let and suppose that one has a -small excess

for some ball compactly contained in .Then, in an appropriate system of coordinates, is the graph of a smooth (and even real-analytic) function.

This theorem is sometimes called a *-regularity theorem* because it provides smoothness (of near ) whenever the excess is below a certain threshold . For this reason, we will refer to it as De Giorgi’s -regularity theorem in what follows.

Since -almost every is a Lebesgue point of the approximate unit normal , an immediate consequence of De Giorgi’s -regularity theorem is the following restriction on the size of the singular set:

Corollary 7 (De Giorgi)If locally minimizes the perimeter in an open set , then is almost everywhere regular in , i.e.,

The proof of De Giorgi’s -regularity theorem is based on the so-called *excess decay lemma*:

Lemma 8 (Excess decay lemma)Let be a locally perimeter minimizing set (in some open set ). Then,For each , there exists a (small) dimensional constant with the following property.

whenever and .

In fact, the excess decay lemma is the starting point of an iteration scheme of the *open* condition in showing that the approximate unit normal at scale is Hölder continuous on *uniformly* on the scale .

From this uniform Hölder continuity property near with , one can show that, for some (and, actually, for all ), the set is the graph (in an adequate system of coordinates) of a function in a neighborhood of .

Since minimizes the perimeter (locally in ), the function is a weak solution to the minimal surfaces equation

In particular, we can invoke the regularity theory of quasilinear elliptic PDEs to conclude that the weak solution of the previous equation is necessarily *smooth* (real-analytic).

In summary, we just saw the sketch of proof of DeGiorgi’s -regularity theorem *modulo* the excess decay lemma.

The proof of the excess decay lemma is by contradiction (and it uses an idea that is helpful for the study of Plateau problem in higher codimensions).

We start with the situation where the excess is very small (), but the excess does not decay as expected. After performing some scalings and rotations (to “blow-up” near ), one obtains a family , , of finite perimeter sets contained in the unit ball of such that and the corresponding approximate unit normals deviate very little from an arbitrarily fixed direction (independent of ).

This means that can be approximated by the graph of a function on the unit ball of with a very small Lipschitz constant. By using this information to linearize the area functional, one gets that the perimeter of has an expansion of the form

Since are obtained from by scalings and rotations, also minimizes the perimeter. By combining this fact with the previous expansion of the area functional, one can prove that as and is a *harmonic* function (as minimizes the Dirichlet energy ).

In this way, one gets a contradiction in the limit because any harmonic function satisfies the decay estimate

(where ), a property that is incompatible with our assumption that (and a fortiori ) do not have the expected decay of excess.

** 3.2. Tangent cones and the codimension of singular sets **

Besides the excess, De Giorgi introduced other key tools in the analysis of area-minimizing currents such as the *tangent cone*.

In simple terms, the tangent cone arises from blow-ups of a current near a given point.

More precisely, for and , let us consider the scaling . Given a current , let be defined by . The family of currents corresponds to zooming in at .

Theorem 9If is an area-minimizing current and is an “interior point”, then any weak- limit of the currents (as ) is a cone without boundary, i.e.,

which is locally area-minimizing in , i.e.,

whenever is compactly supported in a bounded open set .

Definition 10Any cone as in the theorem above is called atangent coneto at .

Remark 6An important open problem is theuniquenessof tangent cones.

An important consequence of the theorem above (of existence of tangent cones) and De Giorgi’s -regularity theorem is the following characterization of the regular set of an area-minimizing currents in *codimension one*. Let be a area-minimizing -dimensional current in . Then:

is regular *if and only if* some tangent cone of at is flat

Indeed, this happens because the excess of any flat cone is zero (and, a fortiori, it is below the critical threshold in De Giorgi’s -regularity theorem).

Remark 7As we will see later, this characterization of is particular of the codimension one case. In fact, it is simplyfalsein higher codimensions!

Remark 8Behind the characterization of singular points in terms of non-flat tangent cones, there is a principle of “persistence of singularities”: more precisely, if are singular points of (locally in ) area-minimizing currents , and and as , then .

This intimate relationship between flat cones and regular points (in the codimension case) gives a precise description of the singular set of area-minimizing currents. This fact is the starting point of the proof of the following (optimal) theorem on the codimension of the singular set:

Theorem 11Let be a set of finite perimeter. Assume that locally minimizes perimeter in some open set .

- If , then ;
- If , then is discrete;
- In general, for all , so that the Hausdorff dimension of the singular set is .

The proof of this theorem goes as follows. By the principle of persistence of singularities, the tangent cone at a singular point (obtained the scalings ) is singular at the origin. Furthermore, we saw that is a minimal cone (i.e., is a cone without boundary which locally minimizes area). In other words, a tangent cone at is always a *singular minimal cone*.

By the series of works of De Giorgi, Fleming, Almgren and Simons, there are *no* singular minimal cones when , and there is a singular minimal cone when , namely, Simons’ cone

From this, we deduce that the singular set is empty when (and it might be non-empty when ), so that the first item of the theorem is proved.

The proofs of the two remaining items of the theorem (the discreteness of the singular set when and the estimative of its codimension in general) rely on Federer’s *reduction of dimension argument*.

For the sake of exposition, we will illustrate this argument by proving just the discreteness of the singular set when . If is not discrete for some , we would have a sequence converging to (by the principle of persistence of singularities).

By making a blowup of (on ) at scales , we obtain a tangent cone which is singular at the origin *and* at some point in the unit sphere of (again by the principle of persistence of singularities).

Since is a cone, the half-line between and is contained in . By doing a *new* blowup at the middle point of the segment between and , we obtain a new tangent cone of the form (where the factor comes from the contribution in the limit of to the blowup) such that would be a singular minimal cone in , a contradiction with the fact that there are no singular minimal cones in when .

At this point, we dispose of sharp regularity results for solutions of the generalized Plateau problem in codimension (area-minimizing -dimensional currents in ), so that it is time to move to the case of higher codimensions.

**4. Plateau’s problem in higher codimensions **

The regularity theory of area-minimizing -dimensional currents in when is *substantially* more involved than its codimension one counterpart due to the following *new phenomena*:

- the presence of flat tangent cones at a point does
*not*imply its regularity; - the presence of
*branch points*in the singular set; - the necessity of
*non-homogenous*blowups (i.e., push-forwards by are no longer sufficient to understand the local geometry of area-minimizing currents).

In fact, these difficulties already appear in the context of Example 2, i.e., the singular -disk

with boundary .

** 4.1. Examples of center manifolds and multivalued functions **

Note that is locally area-minimizing: indeed, as we already mentioned in Example 2, this is a direct consequence of the calibration method.

More precisely, given an integral rectifiable current (with orientation ), we say that is *calibrated* by a smooth closed differential -form on an open subset with for all whenever

for -almost every .

The fundamental (and elementary) remark based on Stokes formula is that calibrated currents are locally area-minimizing. Furthermore, the Wirtinger inequality says that any complex submanifold of complex dimension induces a current that is calibrated by

In particular, these facts imply that is locally area-minimizing.

Observe that this shows that Almgren’s theorem 1 is an optimal solution to Plateau’s problem in codimension : in fact, is a integral rectifiable locally area-minimizing -dimensional current in with a singular set of Hausdorff dimension .

By considering the scalings

we see that the tangent cone to at (i.e., the weak- limit of as ) is the integral rectifiable current

associated to the horizontal plane with *multiplicity two*. Geometrically, this means that, after scaling (“zooming in near the origin”), the *two branches* of (corresponding to the two determinations of , i.e., square-root of , near ) merge together into the plane .

It is worth to notice that the tangent cone of at is flat, but, *nevertheless*, is a *singular point* of due to its *branch nature*. In other words, contrary to the case of Plateau’s problem in codimension , it is *no* longer true in *higher codimensions* that flat tangent cones are associated only to regular points.

Remark 9If we have someextrainformation on flat tangent cones (e.g., if we knowin advancethat its multiplicity is one), then we can still ensure that the associated point is regular.

An important point in the example of (near the origin) is that it behaves *differently* along the and coordinates. This suggest that we should replace the *homogenous* scalings by *non-homogenous* scalings taking into account the distinct behaviors of the and variables along . Moreover, by analogy with the case of Plateau’s problem in codimension , we want a non-homogenous scalings such that the limit objects (multivalued functions) are *harmonic*.

As it turns out, the *correct* non-homogenous scaling of near the origin is . In fact, the main point here is that this scaling *fixes* (that is, ) because the functions and corresponding to the determinations of the square-root of (i.e., ) are *already* harmonic functions.

In general, it is an important problem to construct non-homogenous scalings leading to *non-trivial* limit objects (blowups), and, in fact, almost half of Almgren’s work is dedicated to address this issue.

In order to get a grasp on the tools introduced by Almgren to overcome these difficulties, let us consider another example. Let

By the calibration method, we have that is locally area-minimizing.

Also, it is not hard to check that is singular at the origin. If one tries to play with non-homogenous scalings , , then it is possible to verify that the *sole* scaling producing an interesting (non-trivial and “harmonic”) blowup is

A quick calculation reveals that converges (as ) to the current induced by the *smooth* complex curve

In particular, this non-homogenous scaling of near the origin leads to a *non-flat* limit object (which is certainly not a cone)!

This suggests that we should *forget* the idea of getting tangent cones as the blowup (limit object) under non-homogenous scalings.

In fact, the main point of the example of is that the induced current has a “regular part” given by the smooth curve and a “singular branching” part coming from the determination of the square-root of . This means that the parametrization of the current with the *flat* coordinates is not adequate: instead, it is a better idea to parametrize in terms of the smooth complex curve , i.e., we have a complex parameter for the smooth complex curve (“regular part of ”), and we have two functions and with parametrizing the “singular branching part of ”.

This decomposition led Almgren to introduce the notions of *center manifolds* (“regular parts”) and *multivalued harmonic functions* (“singular branching part”). In the case of , the curve is the center manifold, and the multivalued function is the multivalued harmonic function.

Almgren’s construction of center manifold (via adequate non-homogenous scalings) is very delicate. This construction was simplified by DeLellis-Spadaro, but it still consists a complicated part of the proof of Almgren’s theorem 1.

For this reason, let’s assume that we already dispose of adequate non-homogenous scalings and center manifolds, and we consider now the description of the “singular branching” using multivalued harmonic functions.

Very roughly speaking, multivalued functions are obtained by a sort of averaging procedure of the branches of the current over the center manifold.

Remark 10It is important to ensure that the “order of contact” of the branches with the center manifold is not infinity (otherwise one would get a trivial blowup with this procedure). In this direction, Almgren introduced a certain monotonicity formula allowing him to prove that the “order of contact” is always finite.

More concretely, a multivalued function (over the center manifold ) is defined as follows. We consider (“number of branches”), and we let

be the space of unordered -tuples of points in . Here, is the equivalent relation for all and is a permutation of .

In this setting, a –*valued function* is simply a function .

Note that is a metric space. For example, this space possesses the (Wasserstein) metric

This metric space structure on allows us to talk about *Lipschitz* -valued functions, but this is not satisfactory for our purposes: recall that we want to make sense of *harmonic* multivalued functions!

** 4.2. Two approaches to harmonic multivalued functions **

Almgren proposed the following *extrinsic* approach for the definition of -valued harmonic functions. First, one tries to *isometrically* embed into some Euclidean space . Then, one defines Sobolev spaces, Dirichlet energy and harmonicity (i.e., minimizers of the Dirichlet energy) of -valued functions with the aid of the nice structures of : for example, the Sobolev space is defined as the space of functions (in the usual sense) such that for almost every .

As it turns out, these definitions work well when is isometrically embedded in in such a way that it is bi-Lipschitz equivalent to a *Lipschitz retract* of . This led Almgren to prove that, in fact, this is always the case for some .

In their proof of Almgren’s theorem, DeLellis-Spadaro proposed an alternative *intrinsic* approach for the definition of -valued harmonic functions: the idea is to rely only on the metric structure of (without making any mention to ambient spaces like ). More precisely, we say that belongs to the Sobolev space if there are , such that

- for each , the function belongs to the (usual) Sobolev space ;
- for each and , for almost every .

In other words, we measure the (Sobolev) regularity of by testing the Sobolev regularity of the real-valued functions on obtained by measuring the -distance of the values of to arbitrary points .

Remark 11One can show that there are (minimal) functions such that for any -functions satisfying the inequalities above.

In this context, the *Dirichlet energy* of is

where . Moreover, we say that a -valued function is *harmonic* if minimizes the Dirichlet energy (with Dirichlet or Neumann boundary condition).

In any event, the notion of harmonic -valued function plays a key role in the proof of Almgren’s theorem 1 because the graphs of such functions (describing the “singular branching parts” of area-minimizing currents) have the following regularity property:

Theorem 12 (Almgren-DeLellis-Spadaro)Then, there exists minimizing the Dirichlet energy in the class of with . Furthermore:Let be a bounded open subset with Lipschitz boundary (e.g., a piece of the center manifold ), and .

- any such is locally -Hölder continuous for a certain , and is locally for some ;
- there exists a closed (“singular”) subset with Hausdorff dimension such that the graph
of outside is a smooth embedded -dimensional submanifold of .

At this point, we are ready to give a global (rough) description of the proof of Almgren’s theorem 1 (assuming the statement of Theorem 12).

** 4.3. Sketch of proof of Almgren’s theorem **

Closing this post, we note that the general scheme is similar to De Giorgi’s regularity theory in codimension one, even though the details are obviously different (and much more technical):

- one starts with a locally area-minimizing integral rectifiable current and one makes a blowup near a point using non-homogenous scalings;
- in this way, one obtains a center manifold (describing the regular part of the current) and Lipschitz multivalued functions (describing the singular branching part of the current) providing approximations to the branches of the scalings of the area-minimizing current;
- one shows that the quality of approximation of these Lipschitz multivalued functions depend in a
*superlinear*way on a certain (variant of De Giorgi’s)*excess*; by combining this fact with an expansion of Dirichlet’s energy and (and Almgren’s monotonicity formula for the control of the “order of contact”), one obtains a -valued harmonic function as the limit of these Lipschitz multivalued functions; - finally, one applies the regularity estimate of Theorem 12 to this -valued harmonic function to conclude the proof of Theorem 1 (i.e., to deduce that the singular set of an area-minimizing integral rectifiable -dimensional current has Hausdorff dimension ).

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This event will consist into three talks by Giovanni Forni, Jean-Christophe Yoccoz and Anton Zorich, and a tentative schedule is available here. Also, it is likely that these talks will be recorded, and, in this case, I plan to update this (very short) post by providing a link for the eventual videos of these lectures.

Please note that any interested person can attend this event (as there are no inscription fees). On the other hand, since our budget is very limited, unfortunately Luca and I can not offer any sort of financial help (with local and/or travel expenses) to potential participants. In particular, we would ask you to use your own grants to support your eventual participation in “Journée Surfaces Plates”.

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**Disclaimer.** All errors, mistakes or misattributions are my entire responsibility.

**1. Introduction **

Given a Riemannian -dimensional manifold , one can often study its Geometry by analyzing adequate smooth real functions on (such as scalar curvature). One of the techniques used to get some information about is the following observation (“baby maximum principle”): if has a local maximum at a point , then we dispose of

- a
*first order*information: the gradient of at vanishes; and - a
*second order*information: the Hessian of at has a sign (namely, it is negative definite).

In order to extract *more* information from this technique, one can appeal to the so-called *doubling of variables method*: instead of studying , one investigates the local maxima of a “well-chosen” function on the *double* of variables (e.g., ). In this way, we have *new* constraints because the gradient and Hessian of depend on more variables than those of .

This idea of doubling the variables goes back to Kruzkov who used it to estimate the modulus of continuity of the derivative of solutions of a non-linear parabolic PDE (in one space dimension). In this post we shall see how this idea was ingeniously employed by Andrews and Clutterbuck (2011) and Brendle (2013) in two recent important works.

We start with the statement of Andrews-Clutterbuck theorem:

Theorem 1 (Andrews-Clutterbuck)Recall that the spectrum of with respect to Dirichlet condition on the boundary consists of a discrete set of eigenvalues of the form:Let be a convex domain of diameter . Consider the Schrödinger operator where is the Laplacian operator and is the operator induced by the multiplication by a convex function .In this setting, the

fundamental gapof is bounded from below by

Remark 1This theorem is sharp: when and (by Fourier analysis). In other terms, Andrews-Clutterbuck theorem is anoptimalcomparison theorem between the fundamental gap of general Schrödinger operators with the one-dimensional case.

Next, we state Brendle’s theorem:

Theorem 2 (Brendle)A minimal torus inside the round sphere is isometric to Clifford torus .

The sketches of proof of these results are presented in the next two Sections. For now, let us close this introductory section by explaining some of the motivations of these theorems.

** 1.1. The context of Andrews-Clutterbuck theorem **

The interest of the fundamental gap comes from the fact that it helps in the description of the long-term behavior of non-negative non-trivial solutions of the heat equation

with on . More precisely, one has that

where

- is an adequate constant,
- is the
*ground state*of , i.e., , on , on and is normalized so that , and - denotes (as usual) a quantity bounded from above by for some constant and all .

The theorem of Andrews-Clutterbuck answers positively a conjecture of Yau and Ashbaugh-Benguria. This conjecture was based on a series of works in Mathematics and Physics: from the mathematical side, van den Berg observed during his study of the behavior of spectral functions in big convex domains (modeling Bose-Einstein condensation) that for the free Laplacian () on several convex domains. After that, Singer-Wong-Yau-Yau proved that

and Yu-Zhong improved this result by showing that

Furthermore, some particular cases of Andrews-Clutterbuck were previously known: for instance, Lavine proved the one-dimensional case , and other authors studied the cases of convex domains with some (axial and/or rotational) symmetries in higher dimensions.

** 1.2. The context of Brendle theorem **

The theorem of Brendle answers affirmatively a Lawson’s conjecture.

Lawson arrived at this conjecture after proving (in this paper here) that *every* compact oriented surface without boundary can be *minimally embedded* in .

Remark 2The analog of Lawson’s theorem is completely false in : using the maximum principle, one can show that there arenoimmersed compact minimal surfaces in .

Moreover, Lawson (in the same paper loc. cit.) showed that, if the genus of is *not* prime, then admits two *non-isometric* minimal embeddings in .

On the other hand, Lawson’s construction in the case of genus produces *only* the Clifford torus (up to isometries). Nevertheless, Lawson proved (in this paper here) that if is a minimal torus, then there exists a *diffeomorphism* taking to the Clifford torus : in other terms, there is *no* knotted minimal torus in !

In this context, Lawson was led to conjecture that this diffeomorphism could be taken to be an *isometry*, an assertion that was confirmed by Brendle.

**2. Proof of Andrews-Clutterbuck theorem **

One of the key points of Andrews-Clutterbuck argument is an *improvement* of a theorem of Brascamp-Lieb. More precisely, the Brascamp-Lieb theorem ensures, in the context of Theorem 1, the log-concavity of the the ground state of (i.e., the logarithm is a concave function). In this setting, a fundamental ingredient in Andrews-Clutterbuck proof of Theorem 1 is a *quantitative* statement about the log-concavity of .

Before discussing Andrews-Clutterbuck’s improvement of Brascamp-Lieb theorem, let us quickly review Korevaar’s proof of Brascamp-Lieb theorem as an excuse to introduce a first concrete instance of the doubling of variables method.

** 2.1. A sketch of Korevaar’s proof of Brascamp-Lieb theorem **

We want to show that is log-concave. For this sake, we can assume that the domain and the potential are *strictly convex*. Indeed, this is so because and are convex, so that they can be approximated by strictly convex objects, and, furthermore, it can be shown that the ground state varies *continuously* under deformations of and .

By definition, is concave if and only if the function

on the double of variables is non-positive.

We divide the proof of the fact that for all into two parts.

First, we claim that . In fact:

- If with , then because (i.e., ) on and on . Here, we used that .
- If with , one exploits the strict convexity of to say that, near , the ground state “looks like” the distance to the boundary , so that is a concave function near .

Next, once we dispose of the fact that , the proof of the log-concavity of will be complete if we show that at any local maximum .

In this direction, we use the baby maximum principle. If is a local maximum of , then vanishes to the first order at , i.e., . Thus, if denoting by , we deduce from the definition of and the equation that

Moreover, by varying in the direction of a small vector , we get a function

possessing a local maximum at . Therefore, the Laplacian of this function at is non-positive, i.e.,

Now, a simple calculation reveals that the Laplacian of satisfies the equation

because is the ground state of (i.e., ). Combining this equation with (1) and (2), we conclude that

Since is strictly convex, this inequality implies that , and, *a fortiori*, , as we wanted to prove. This completes the sketch of Korevaar’s proof of Brascamp-Lieb theorem.

** 2.2. An improvement of Brascamp-Lieb’s theorem **

The improvement of Andrews-Clutterbuck of the Brascamp-Lieb theorem consists of the following estimate of the modulus of continuity of the derivative of :

This estimate provides new important informations beyond the statement of Brascamp-Lieb theorem: for example, when , the right-hand side of the inequality goes to (which is much better than simply knowing that it is non-positive).

The proof of this estimate is somewhat complicated: it involves a combination of the doubling of variables method, a comparison argument with the one-dimensional case and the study of parabolic PDEs.

For this reason, by following Carron’s talk, we will skip the proof of this estimate, and we will now discuss how this estimate can be used to get lower bound on the fundamental gap in Theorem 1. In this direction, we will follow an approach proposed by Lei Ni (which is not exactly the original argument of Andrews-Clutterbuck).

** 2.3. End of the (sketch of) proof of Theorem 1 **

We consider the eigenfunction with , , , and on .

The quotient is closely related to the fundamental gap : more precisely, the function verifies

and on (where is the unit outward normal to ).

The previous method of Singer-Wong-Yau-Yau consisted of studying first two derivatives of the function

at its local maximum points, extract an inequality, and obtain a (non-optimal) lower bound on by integration of this inequality (together with the fact that satisfies (4)).

The approach proposed by Andrews-Clutterbuck consists in studying the oscillations of , i.e., we compare with the one-dimensional model (on an interval) by means of the function:

where and . (This choice of avoids “boundary effects”).

We want to use the doubling of variables method, i.e., the baby maximum principle applied to . For this sake, we introduce the quantity:

One can show that the strict convexity of implies that is attained on *or* in the interior of .

Remark 3We have not defined on , but a first order expansion says that it is natural to pose

In both cases, the study of the first two derivatives of at a maximum point *and* the improvement (3) of Brascamp-Lieb theorem imply that

Of course, since is arbitrary, this proves that . Hence, the proof of Theorem 1 is complete once we prove (5).

For the sake of exposition, we show the validity of (5) *only* in the case that is attained at , i.e., for some : indeed, the other case is similar (in some sense) to this one.

If is attained at ,, then is a maximum for , so that

Of course, this inequality is a good starting point to study (via (4)), but it is only a *partial* information obtained by letting vary *only* along !

If we vary along the *transverse* direction by considering where is a small vector, we obtain from (and the baby maximum principle) that

which is certainly a *better* estimate than the previous one.

In other words, we got an extra (better) information on thanks to the doubling of variables method applied to !

By differentiating the equation (4), and then applying (6) to the resulting PDE, we deduce that

Now, we notice that the Andrews-Clutterbuck improvement (3) of Brascamp-Lieb theorem says (among other things) that . By plugging this into the previous inequality, we conclude that

Since is a non-constant function (and ), we have from this inequality that

This proves (5) when is attained at , as desired.

**3. Proof of Brendle theorem **

Let be a minimal torus inside the round -sphere . Denote by a choice of unit normal to .

The second fundamental form is the Hessian at of the function (from to ) whose graph over is (locally) equal to . In particular, is a symmetric quadratic form, and, hence, can be diagonalized. The (real) eigenvalues of are called principal curvatures of at .

By definition, is minimal if and only if the trace of vanishes (for all ). In other words, the eigenvalues of are when is minimal.

For later use, we recall the following three facts:

- Lawson proved that when is a minimal torus in . (Of course, this result strongly uses that has genus , and, indeed, it is completely
*false*for other genera) - The minimality of imposes a constraint on known as
*Simon’s formula*. In our setting, this means that the principal curvature verifies the following PDE: - Lawson also showed that if is constant, then the minimal torus is isometric to Clifford’s torus .

The last item above says that our objective is very clear: in order to prove Brendle’s theorem 2, we have to show that is constant.

By Gauss-Bonnet theorem, we have that , i.e., equals to in average. From this point, a natural strategy would be to combine this information with Simons’ equation (and some maximum principles) to show that . Unfortunately, this idea does *not* work mainly because of the (negative) sign of the term in Simons formula.

At this point, Brendle introduces the function

(Note that, since , , and, thus, .)

The *geometrical* meaning of is the following. The quantity is the biggest radius such that stays outside a ball of radius tangent to at .

In other terms, stays outside of the oscullating balls of radius , and , and are mutually tangent at . From this fact, it is possible to check that

This means that the *global* information (curvature of oscullating balls) controls the *local* information (principal curvature).

We *affirm* that the inequality implies that satisfies the following version of Simons formula

in the sense of viscosity. For the sake of exposition, let us prove that satisfies this inequality when : the general case ( is a viscosity solution when is not smooth) follows by a simple modification of the argument below.

Up to changing our choice of unit normal , we can write . Let us apply the doubling of variables method by considering the function

Given a point , we have two possibilities: either or .

In the first case (), since , one has (from the baby maximum principle) that and . By plugging this into Simons formula (7), we deduce that

In the second case , we have that there exists , such that . Since , we get that . Geometrically, this condition means that stays outside the ball and is tangent to at and . In particular, this implies that the tangent planes and are symmetric with respect to the mediator hyperplane of the segment between and . By exploiting this symmetry, Brendle chose good coordinate systems on and leading him to the following inequality

with after *seven* pages of calculations in his paper! Since , we have the good sign to conclude from this estimate that

as it was claimed.

Once we know that is a (viscosity) solution of (8), we can use the maximum principles, the inequality and the Simons formula (7) for to obtain that

where is a constant.

We *claim* that this implies that is constant, so that the proof of Brendle theorem would be complete (by Lawson’s result quoted right after Simons formula (7)). Indeed, we have again two cases: either or .

In the first situation, since the oscullating balls with have the same principal curvature of at , a third order expansion of (the graph of) at reveals that for all , so that is constant.

In the second situation with , we look again at the inequality

showed above (under the assumption , which is our current situation at *all* since , ).

Because satisfies Simons formula (7) and with a constant, we get that the first term of the previous inequality vanishes. In particular, we deduce that

This means that for all (since and ), so that is constant, as it was claimed.

This completes the proof of Brendle’s theorem and, consequently, the discussion of this post.

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