with is encoded by the so-called *Lagrange spectrum* .

In a similar vein, the Markov spectrum encodes best constants for a Diophantine problem involving indefinite binary quadratic real forms.

These spectra were first studied in a systematic way by A. Markov in 1880, and, since then, their structures attracted the attention of several mathematicians (including Hurwitz, Perron, etc.).

Among the basic properties of these spectra, it is worth mentioning that are closed subsets of the real line. Moreover, the works of Markov from 1880 and Hall from 1947 imply that

is a increasing sequence of quadratic surds converging to , and

On the other hand, it took some time to decide whether . Indeed, Freiman proved in 1968 that by exhibiting a countable (infinite) collection of isolated points in . After that, Freiman constructed in 1973 an element of which was shown to be a non-isolated point of by Flahive in 1977.

A common feature of these examples of elements in is the fact that they occur before In 1975, Cusick conjectured that there were no elements in beyond .

In our preprint uploaded to arXiv a couple of days ago, Gugu and I provide the following negative answer to Cusick’s conjecture:

Below the fold, we give an outline of the proof of this theorem.

Remark 1The basic reference for this post is the classical book of Cusick and Flahive.

**1. Description of the key ideas**

Recall that and , where

and is the usual continued fraction expansion.

We consider the finite word of odd length: this is a *non* semi-symmetric word in the sense of Flahive (i.e., it can *not* be decomposed into two palindromes).

Remark 2The examples constructed by Freiman of elements in were based on two non semi-symmetric words, and Flahive showed in her paper that “an element of is often associated to non semi-symmetric words”.

The periodic sequence obtained by infinite concatenation of has Markov value

If we try to glue the word on the *right* of , we get a new sequence with Markov value

On the other hand, if we try to glue the word on the *left* of in a way to obtain the smallest possible change in the Markov value, then the best choice is the sequence whose Markov value is

Hence, the cost of gluing to the left of is always higher than the cost of . This indicates that the Markov value of *doesn’t* belong to because any attempt to reproduce as the of for some sequence would force the appearance of large chunks of at arbitrarily large positions in , so that would contain a subword (which is essentially) arbitrarily close to and, thus, its Markov value would be *at least* the Markov value of , a contradiction (since ).

Closing this section, we remind for later use the following standard comparison lemma for continued fractions.

**2. “Big” words and self-replication**

The following lemma provides a list of words whose appearance in a sequence forces its Markov value to be

Lemma 3If contain any of the words

- (1)
- (2)
- (3)
- (4)
- (5)
- (6)
- (7)
- (8)
- (9)
- (10)
- (11)
- (12)
- (13)
- (14) or
- (15) or
- (16) or
- (17)
- (18) or
- (19)
- (20)
- (21)
- (22)
- (23)

and is the position in asterisk, then

for some .

*Proof:* We prove this lemma by straightforward calculations using the standard comparison Lemma 2.

We verify the items in their order of appearance: for example, , , etc.

Sometimes, we need to use the previous items ((1), (2) and/or (5)) and the assumption that for all to get the desired conclusion for a given item: for instance, by (1) and the fact that , we obtain (3):

See our original article for more details or this Mathematica notebook here.

A direct consequence of this lemma is the fact that the word

must extend as

whenever the Markov value is .

Corollary 4Let where is the position in asterisk. If for all , then extends as

and the vicinity of the position is .

*Proof:* By Lemma 3 (3) and (19), extends to the left as

By Lemma 3 (5) and (20), continues to the right as

By Lemma 3 (6) and (7), extends to the right as

By Lemma 3 (1) and (8), continues to the right as

By Lemma 3 (1) and (21), extends to the left as

By Lemma 3 (10) and (22), continues to the left as

By Lemma 3 (11) and (23), extends to the left as

By Lemma 3 (15), continues to the left as

By Lemma 3 (1), (2) and (4), extends to the left as

This completes the proof.

**3. “Small” words and local uniqueness**

The next lemma gives a list of words associated to positions where a Markov value is *not* reached.

Lemma 5If contain any of the words

- (24)
- (25)
- (26)
- (27) or
- (28)
- (29) or
- (30)
- (31)
- (32) or
- (33)
- (34) or
- (35)
- (36)
- (37) or
- (38) or

and is the position in asterisk, then

*Proof:* The proof of this lemma is a straightforward calculation based on the standard comparison Lemma 2: see our original article or this Mathematica notebook here for more details.

By putting together Lemmas 3 and 5, we show below that a Markov value between and is necessarily attained at positions whose vicinities are

and for all , then, up to transposition about the th position,

where the asterisk indicates the th position.

*Proof:* By Lemma 3 (1) and (2), and Lemma 5 (24), we have that, up to transposition,

By Lemma 3 (3) and Lemma 5 (25), extends to the right as

By Lemma 3 (1) and (4), extends to the left as

By Lemma 3 (5) and Lemma 5 (26), continues to the right as

By Lemma 5 (27), is forced to extend to the left as

By Lemma 3 (1) and Lemma 5 (28), continues to the left as

By Lemma 3 (6), (7) and (8), is forced to continue on the right as

By Lemma 3 (10) and Lemma 5 (30), extends on the left as

By Lemma 5 (31) and Lemma 3 (1), (2), continues as

or

We claim that the first possibility can’t occur. Indeed, by Lemma 5 (32) and Lemma 3 (1), (2) and (4), this word would extend to the left as

so that its left side would contain the word , a contradiction with Lemma 3 (5). Therefore, we have to analyse the word

By Lemma 3 (11) and Lemma 5 (33), extends on the left as

By Lemma 3 (1) and (13), continues on the right as

By Lemma 3 (15), is forced to extend on the left as

By Lemma 3 (1), (2) and (4), continues on the left as

This ends the proof.

**4. Conclusions**

The first consequence of our previous discussion is the absence of Lagrange spectrum in the interval with extremities and .

Proposition 7

*Proof:* Suppose that , say for some .

By definition, we can select and an increasing sequence such that

for all , and

for each .

By applying Corollary 6 at the positions and Corollary 4 at adequate positions of the form with , we deduce that

a contradiction. (Here denotes the periodic sequence obtained by infinite concatenation of the word .)

Hence, any Markov value in the interval with extremities and does not belong to the Lagrange spectrum. As it turns out, it is not hard to construct a whole Cantor set of such values: in fact, it is not difficult to check that the Markov values of the sequences

form a Cantor set isomorphic to such that

In particular, we deduce that

and the Hausdorff dimension of is equal to the Hausdorff dimension of . This completes the proof of Theorem 1 in view of the recent results of Jenkinson and Pollicott on the Hausdorff dimension of .

]]>The motivation for the work of Irie–Marques–Neves is a famous conjecture of Yau on the abundance of minimal surfaces.

More precisely, Yau conjectured in 1982 that a closed Riemannian -manifold contains infinitely many (smooth) closed immersed minimal surfaces. Despite all the activity around this conjecture, the existence of infinitely many embedded minimal hypersurfaces in manifolds of positive Ricci curvature of low dimensions was only established very recently by Fernando and André.

In their remarkable paper, Irie–Marques–Neves show that Yau’s conjecture is *generically* true in low dimensions by establishing the following *stronger* statement:

Theorem 1Let be a closed manifold of dimension . Then, a generic Riemannian metric on has a lot of minimal hypersurfaces: the union of all of its closed (smooth) embedded minimal hypersurfaces is a dense subset of .

Remark 1The hypothesis of low dimensionality is related to the fact that area-minimizing minimal hypersurfaces in dimensions might exhibit non-trivial singular sets (as it was famously proved by Bombieri–De Giorgi–Guisti), but such a phenomenon does not occur in low dimensions for “min-max” minimal hypersurfaces thanks to the regularity theories of Almgren, Pitts and Schoen–Simon.

The remainder of this post is dedicated to the proof of this theorem and, as usual, all eventual errors/mistakes in what follows are my entire responsibility.

**1. Description of the key ideas**

Let be a closed manifold and be the space of Riemannian metrics on .

Given an open subset , let be the subset of Riemannian metrics possessing a *non-degenerate*closed (smooth) embedded minimal hypersurface passing through . (Here, *non-degenerate* means that all Jacobi fields are trivial.)

It is possible to check that a non-degenerate closed embedded minimal hypersurface in is *persistent*: more concretely, one can use the definition of non-degeneracy and the inverse function theorem to obtain that any Riemannian metric close to possesses an unique closed embedded minimal hypersurface nearby . In particular, every is *open*.

Also, let us observe (for later use) that the non-degeneracy condition is not difficult to obtain:

Proposition 2Let be a closed (smooth) embedded minimal hypersurface in the Riemannian manifold . Then, we can perform conformal perturbations to find a sequence of metrics converging to (in -topology) such that is a non-degenerate minimal hypersurface of for all sufficiently large .

*Proof:* This statement is Proposition 2.3 in Irie–Marques–Neves paper and its proof goes along the following lines.

Fix a bump function supported in a small neighborhood of and coinciding with the square of the distance function nearby .

The metrics are conformal to , and they converge to in the -topology. Furthermore, the features of the distance function imply that is a minimal hypersurface of such that the Jacobi operator acting on normal vector fields verify

for all . In particular, the spectrum of is derived from the spectrum of by translation by . Therefore, doesn’t belong to the spectrum of for all surfficiently large, and, hence, is a non-degenerate minimal hypersurface of for all large .

Coming back to Theorem 1, we affirm that our task is reduced to prove the following statement:

Theorem 3Let be a closed manifold of dimension . Then, for any open subset , one has that is dense in .

In fact, *assuming* Theorem 3, we can easily deduce Theorem 1: if is a countable basis of open subsets of , then Theorem 3 ensures that is a countable intersection of open and dense subsets of the Baire space ; in other terms, is a residual / generic subset of such that any satisfies the conclusions of Theorem 1 (thanks to the definition of and our choice of ).

Remark 2Note that, since is a Baire space, it follows from Baire category theorem that is a dense subset of .

Let us now explain the proof of Theorem 3. Given a neighborhood of a smooth Riemannian metric on a closed manifold , and an open subset , our goal is to show that

For this sake, we apply White’s bumpy metric theorem asserting that we can find such that all closed (smooth) immersed minimal hypersurfaces in are non-degenerate.

If , then we are done. If , then all closed (smooth) embedded minimal hypersurfaces in avoid . In this case, we can *naively* describe the idea of Irie–Marques–Neves to perturb to get as follows:

- we perturb
*only*in to obtain whose volume is*strictly*larger than the volume ; - by the so-called
*Weyl law for the volume spectrum*(conjectured by Gromov and recently proved by Liokumovich–Marques–Neves), the –*widths*of are strictly larger than those of ; - since -widths “count” the minimal hypersurfaces, the previous item implies that
*new*minimal hypersurfaces in were produced; - because coincides with outside , the minimal hypersurfaces of avoiding are the exactly same of ; thus, the new minimal hypersurfaces in mentioned above must intersect , i.e., .

In the sequel, we will explain how a slight *variant* of this scheme completes the proof of Theorem 3.

**2. Increasing the volume of Riemannian metrics**

Let as above. Take a non-negative smooth bump function supported in such that for some .

Consider the family of conformal deformations of . Note that for all .

From now on, we fix once and for all such that for all .

**3. Weyl law for the volume spectrum**

Roughly speaking, the –*width* of a Riemannian manifold is the following min-max quantity. We consider the space of closed hypersurfaces of , and –*sweepouts* of , i.e., is a continuous map from the -dimensional real projective space to which is “homologically non-trivial” and, *a fortiori*, is *not* a constant map.

Remark 3Intuitively, a -sweepout is a non-trivial way of filling with -parameter family of hypersurfaces (which is “similar” to the way the -parameter family of curves fills the round -sphere ).

If we denote by the set of -sweepouts such that no concentration of mass occur (i.e., ), then the -width is morally given by

Remark 4Formally speaking, the definition of -width involves more general objects than the ones presented above: we construct by replacing hypersurfaces by certain -dimensional flat chains modulo two, we allow arbitrary simplicial complexes in place of , etc.: see Irie–Marques–Neves’ paper for more details and/or references.

The -width varies *continuously* with (cf. Lemma 2.1 in Irie–Marques–Neves’ paper). Moreover, it “counts” minimal hypersurfaces (cf. Proposition 2.2 in Irie–Marques–Neves’ paper): if has dimension , then, for each , there is a finite collection of mutually disjoint, closed, smooth, embedded, minimal hypersurfaces in with (stability) indices such that

for some integers . (Here, the stability index is the quantity of negative eigenvalues of the Jacobi operator.)

Furthermore, the asymptotic behavior of is described by Weyl’s law for the volume spectrum (conjectured by Gromov and confirmed by Liokumovich–Marques–Neves): for some *universal* constant , one has

In particular, coming back to the context of Section 2, Weyl’s law for the volume spectrum and the fact that has volume strictly larger than mean that we can select such that the -width is *strictly* larger than the -width of , i.e.,

**4. New minimal hypersurfaces intersecting **

We affirm that there exists such that possesses a closed (smooth) embedded minimal hypersurface passing through .

Otherwise, for each , all closed (smooth) embedded minimal hypersurface in would avoid . Since coincides with outside (by construction), the “counting property of -widths” in equation (1) would imply that

for all .

On the other hand, the fact that is bumpy (in the sense of White’s theorem) permits to conclude that the set is *countable*: indeed, a recent theorem of Sharp implies that the collection of connected, closed, smooth embedded minimal hypersurfaces in with bounded index and volume is finite, so that is countable.

Since the -width depends continuously on the metric, the countability of implies that the function is constant on . In particular, we would have

a contradiction with (2). So, our claim is proved.

At this point, the argument is basically complete: the metric has a closed (smooth) embedded minimal hypersurface passing through ; by Proposition 2, we can perturb (if necessary) in order to get a metric such that is a non-degenerate closed (smooth) embedded minimal hypersurface in , that is, , as desired.

]]>In this post, I will transcript my notes of this talk: as usual, all errors/mistakes are my sole responsibility.

**1. Introduction**

General philosophy behind Zimmer’s program: given a compact manifold (say, the -dimensional sphere), we would like to describe the geometrical and algebraic properties of groups of finite type acting faithfully on ; conversely, given our favorite group of finite type, we want to know the class of compact manifolds on which acts faithfully; in this context, Zimmer’s program proposes some answers to these problems when is a lattice in a Lie group.

More precisely, let be a connected Lie group with finite center whose Lie algebra is semi-simple, and let be a connected maximal split torus of . The dimension of , or equivalently, the dimension of the Lie algebra of , is the so-called *real rank* of , and it is denoted by . Let be a lattice of , i.e., a discrete subgroup such that the quotient has finite Haar measure.

For the sake of concreteness, today we will deal exclusively with the prototypical case of and is subgroup of diagonal matrices in with positive entries.

In this setting, Zimmer’s program offers restrictions on the dimension of compact manifolds admitting non-trivial actions of by smooth diffeomorphisms. In this direction, Aaron Brown, David Fisher and Sebastian Hurtado showed here that

Theorem 1Let be a connected compact manifold. Suppose that the lattice of is uniform (i.e., is compact).If there exists a homomorphism with infinite image, then

As we are going to see below, the proof of this theorem is a beautiful blend of ideas from geometric group theory and dynamical systems.

Before describing the arguments of Brown–Fisher–Hurtado, let us make a few comments of the statement of their theorem.

Remark 1The assumption of compactness of is important: indeed, any countable group acts faithfully by biholomorphisms of a connected non-compact Riemann surface (see the footnote 1 of Cantat’s text for a short proof of this fact).

Remark 2The hypothesis of uniformity of is technical: there is some hope to treat non-uniform lattices and, in fact, Brown–Fisher–Hurtado managed to recently extend their result to the case of .

Remark 3The conclusion is optimal: (and, a fortiori, any lattice of ) acts on the real projective space by projective linear transformations. On the other hand, if one changes , then the inequality can be improved: for example, Brown–Fisher–Hurtado proves that when is the symplectic group of real rank .

Remark 4Concerning the regularity of the elements of , even though one expects similar statements for actions by homeomorphisms, Brown–Fisher–Hurtado deals only with -diffeomorphisms because they need to employ the so-called Pesin theory of non-uniform hyperbolicity.Nevertheless, we shall assume that in the sequel for a technical reason explained later.

Remark 5This theorem is obvious when : indeed, a compact manifold whose group of diffeomorphisms is infinite has dimension .Hence, we can (and do) assume without loss of generality that in what follows.

Remark 6The statement of Brown–Fisher–Hurtado theorem is comparable to Margulis super-rigidity theorem providing a control on the dimension of linear representations of .

**2. Preliminaries**

Fix a Riemannian metric on . Denote by a finite set of generators of . For the sake of convenience, we suppose that contains the identity element and is symmetric (i.e., if and only if ).

The length of a word on the letters of the alphabet is denoted by : in other words, is the distance between and in the Cayley graph of .

The ball of radius is

We say that an action is *feeble* whenever if , there exists such that

for all . (Here, stands for the derivative, and the norm is measured with respect to the Riemannian metric fixed above.) Also, we say that an action is *vigorous* if it is not feeble.

The proof of Theorem 1 is naturally divided into two regimes depending on whether is feeble or vigorous.

**3. Feeble actions**

Our goal in this section is to show that if is *feeble*, then there exists a -invariant Riemannian metric on .

Before proving this claim, let us see why it allows to establish Theorem 1 for feeble actions. The claim implies that is a subgroup of isometries of . Since is compact, Myers–Steenrod theorem says that its group of isometries is a compact Lie group. This permits to apply Margulis super-rigidity theorem and the classical theory of compact Lie groups to get the desired inequality .

Let us now prove the claim.

The first ingredient is a lemma of Fisher–Margulis ensuring that a feeble action is “feeble to all orders”, i.e., for all and , there exists such that

for all .

The second ingredient is provided by the so-called *reinforced property (T)* of Lafforgue. In a nutshell, this property says the following. Given a Hilbert space, denote by the group of continuous linear operators. Let be a parameter. We say that a representation is –*moderate* if there exists such that

for all . Given such a representation , we denote by the set of -invariant vectors. The next statement described the reinforced property (T).

Theorem 2 (Lafforgue, de Laat–de la Salle)Let be a uniform lattice of of . Then, there exist

- (1)
- (2) probability measures on supported on

such that for all -moderate , one has a projection with

Remark 7de la Salle is currently working on extending this result to non-uniform lattices.

We want to explore this theorem to produce the desired invariant Riemannian metric in the claim.

Since any Riemannian metric is a section of , let us consider the action of on induced by .

Denote by the Hilbert space of sections of whose first derivatives are (i.e., is a Sobolev space of type ).

Remark 8Sobolev embedding theorem implies that an element of is when .

Observe that the action of on gives a representation

Take a small parameter, where and are the quantities provided the reinforced property (T). By Theorem 2, we have a projection such that

In other words, is a -invariant, -section of which is the limit of the Riemannian metrics

In particular, is non-negative definite. At this point, our task is reduced to prove that is a Riemannian metric, i.e., for all . For this sake, note that if , then the inequality above would give

for all . On the other hand, the action of is feeble of all orders, so that

Since , we get a contradiction unless , i.e., .

This completes the proof of Theorem 1 for feeble actions.

Remark 9We used that here: indeed, we took sufficiently large to apply Sobolev embedding theorem in order to obtain a -smooth object and we exploited the fact that is feeble of order to conclude that is a Riemannian metric.In the case of actions , one replaces the Hilbert spaces by Banach spaces , and one employs the version of the reinforced property (T) for Banach spaces.

**4. Vigorous actions**

In this section, we assume that is vigorous.

Roughly speaking, we are going to treat the case of vigorous actions by exploring the tension between the vigour of the action (creating non-zero Lyapunov exponents) and Zimmer’s super-rigidity theorem for cocycles (saying that the Lyapunov exponents of the action with respect to any invariant probability measure vanish when ).

Logically, the problem with this strategy is the fact that is not amenable, so that the existence of invariant probability measures (required by Zimmer’s super-rigidity theorem) is far from being automatic. In particular, this partly explains why the first versions of Zimmer’s program dealt exclusively with actions of by *volume-preserving* diffeomorphisms of . Also, even if we disposed of invariant probability measures, their supports could be very “thin”, so that their generic points would not “feel” the vigour of the action (and hence no contradiction could be derived).

Anyhow, we will discuss how to overcome the difficulties in the previous paragraph: we shall use the vigour of the action in order to construct an invariant probability measure with some positive Lyapunov exponent, so that the desired conclusion will follow from Zimmer’s super-rigidity.

**4.1. Suspensions**

We start by replacing the action of by a `cousin’ action of . More concretely, consider the product space . Note that acts on via

and acts on via

In particular, acts on the space (because the actions of and commute).

Observe that the action of on is a suspension of the action of on with respect to the natural projection .

We denote by is the vertical tangent bundle (i.e., the tangent space to the fibers of ). Let be the restriction of the derivative of to . Given and a -invariant probability measure on , we define

the maximal vertical Lyapunov exponent of with respect to .

Remark 10For each fixed and , the quantity is a continuous function of . Therefore, for each fixed , the Lyapunov exponent is a upper semi-continuous function of .

Our goal is to exhibit a probability measure on which is -invariant and possessing a positive Lyapunov exponent, i.e.,

for some .

**4.2. -invariant measures**

The first step towards our goal consists in constructing a probability measure on which is -invariant, -ergodic and possessing a positive Lyapunov exponent in the sense that

for some .

For this sake, we recall that a vigorous action has the property that for some and for a sequence , , one has

By Cartan’s decomposition , where is a maximal compact subgroup. Thus, we can write . By compactness, is uniformly bounded for all , so that

where , . (Here we used that and the size of are comparable [because is compact].)

By extracting a subsequence of , we can assume that goes to infinity in a fixed direction with a fixed speed. In particular, this allows us to replace by the iterates of a single element , i.e., we found an element of with

If we take such that , then it is not hard to see that the sequence of probability measures

accumulate into some -invariant probability measure with

Since is amenable and commutes with , we can replace by an -invariant probability measure (by taking averages along F{\o}lner sequences) whose Lyapunov exponent is positive (thanks to the upper semi-continuity property mentioned in Remark 10). Finally, by taking an appropriate ergodic component, we can also assume that is -ergodic.

**4.3. Ratner theory and higher rank groups**

We affirm that the probability measure constructed above can be chosen so that its projection to is

Indeed, the assumption that implies that contains unipotent subgroups commuting with . Since an unipotent subgroup is amenable, we can repeat the argument of the previous subsection (with replaced by such unipotents) to get an *additional* invariance, i.e., we can assume that is invariant under *and* some unipotent subgroup. At this point, Ratner’s theory permits to control the projection and, in particular, to assert that is the Haar measure.

**4.4. Entropy argument**

Let us now show that described above is -invariant on *if* . Note that this completes the proof of Theorem 1 for vigorous actions because the positivity of the Lyapunov exponent contradicts Zimmer’s super-rigidity theorem *unless* .

The vertical Lyapunov exponents of the elements of with respect to define *linear (Lyapunov) forms*

The total number of linear forms (counting multiplicities) is . Since we are assuming that , there exists with

for all .

Let . Recall that the Lyapunov forms associated to the action of on are the trivial form and the roots of . Thus, Pesin entropy formula says that the entropy of the action of on *coincides* with the sum of positive Lyapunov exponent:

(where is the root space of ).

On the other hand, Margulis–Ruelle inequality says that the entropy of action of on is *bounded* by the sum of positive Lyapunov exponents. Since the vertical Lyapunov exponents of vanish, we conclude that

Because projects onto , we also have

In summary, we obtain that Margulis–Ruelle inequality is actually an *equality*. By the invariance theorem of Ledrappier–Young, we derive that is invariant by for all roots with .

By reversing the time (i.e., replacing by ) in the previous argument, we also obtain that is invariant by for all roots with .

Since is the smallest group containing all with (as is a simple Lie group), we conclude the desired -invariance of .

This ends the proof of Theorem 1.

]]>My general plan is to follow the same steps by Jean-Christophe when he became the responsible for Michel Herman archives, namely, I will make available at this webpage here all unpublished texts after selecting and revising them together with Jean-Christophe’s friends.

So far, the webpage dedicated to Jean-Christophe’s archives contains only an original text (circa 1986), a latex version of this text (typed by Alain Albouy, Alain Chenchiner, and myself), and some lecture notes taken by Alain Chenchiner of a talk by Jean-Christophe on the central configurations for the planar four-body problem.

Nevertheless, I hope that this webpage will be regularly updated in the forthcoming years: indeed, Jean-Christophe’s archives takes all cabinets and some corners of an entire office, and, thus, there is more than enough material to keep his friends occupied for some time.

Closing this extremely short post, let me take the opportunity to announce also that Gazette des Mathématiciens (published by the French Mathematical Society) plans to publish in April 2018 a special volume (edited by P. Berger, S. Crovisier, P. Le Calvez and myself) dedicated to several aspects of Jean-Christophe’s mathematical life.

]]>

The key *dynamical* idea to give upper bounds on is to show that any sufficiently large element is realized by a sequence whose past or future dynamics lies in the *gaps* of an appropriate horseshoe.

*Qualitately* speaking, this idea is explained by the following lemma.

Lemma 1Fix a horseshoe of a surface diffeomorphism and a height function. For simplicity, let us denote the orbits of by . Denote by

the corresponding Markov and Lagrange spectra.Let a subhorseshoe of and set

where is the Markov value of . Consider such that , and denote by a point with

Then, either or (where and denote the and limit sets of the orbit of ).

*Proof:* By contradiction, suppose that and .

Since and , we can select and such that for all , and . Also, the definitions allow us to take and such that and .

Fix with dense orbit and consider pieces of the orbit of with and .

Consider the pseudo-orbits . By the shadowing lemma, we obtain a sequence of periodic orbits accumulating whose Markov values converge to . In particular, , a contradiction.

In simple terms, this lemma says that an element with is associated to an orbit whose past dynamics (described by ) or future dynamics (described by ) avoids . Thus, there exists such that either the piece of past orbit or the piece of future orbit avoids a neighborhood of in (i.e., one of these pieces of orbit lives in the gaps of ).

Remark 1As it turns out, this qualitative lemma is not sufficient for our purposes and, for this reason, Gugu and I end up using a quantitative version of this lemma (called Lemma 3.1) in our paper.

Once we got some constraints on the dynamics of orbits generating elements of , our strategy to estimate consists in careful choices of and .

For the sake of exposition, let us explain how our strategy yields some bounds for .

Perron proved that any has the form where .

Consider the subhorseshoe (of sequences formed by concatenations of two consecutive 1’s and two consecutive 2’s).

By applying (a quantitative version of) Lemma 1 (cf. Remark 1), one concludes that if , then the past or future dynamics of lives in the *gaps* of .

This means that, up to replacing by , for all sufficiently large:

- either there is an
*unique*extension of giving a sequence whose Markov value in ; - or there are two continuations and of so that the interval is
*disjoint*from the Cantor setassociated to .

(Here, denotes continued fraction expansions.)

Note that this dichotomy imposes *severe* restrictions on the future of because there are not many ways to build sequences associated to . More precisely, we claim that at each sufficiently large step ,

- either we get a forced continuation ;
- or our possible continuations are and .

Indeed, suppose that we have two possible continuations and . If denotes the interval of numbers in whose continued fraction expansion starts by , then the intervals , appear in the following order on the real line:

Thus:

- the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect ; - the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect

so that our continuations are and . Now, we observe that the intervals and , , appear in the following order on the real line:

Hence:

- the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect ; - the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect

so that our claim is proved.

This claim allows us to bound the Hausdorff dimension of

In fact, the claim says that we refine the natural cover of by the intervals by replacing it by a “forced” or by the couple of intervals

Therefore, it follows from the definition of Hausdorff dimension that

Because we can assume that with and , our discussion so far can be summarized by following proposition:

Proposition 2is contained in the arithmetic sum

where for any parameter satisfying (1).

Since the arithmetic sum is the projection , , of the product set , this proposition implies the following result:

Corollary 3where satisfies (1).

The Hausdorff dimension of was computed with high accuracy by Hensley among other authors: one has . In particular,

where verifies (1).

Closing this post, let us show that (1) holds for and, consequently,

For this sake, recall that

where is the denominator of .

Hence, if we set

then the recurrence formula implies that

where .

Because and for all , we have

This completes the argument because .

]]>In this context, the entries of the continued fraction expansion of an irrational number are related to *cusp excursions* of typical geodesics in the modular surface (i.e., visits to regions for large).

By exploiting this relationship, Vaibhav Gadre analysed cusp excursions on the modular surface to obtain a proof of the following theorem originally due to Diamond–Vaaler:

Theorem 1For Lebesgue almost every , one has

Furthermore, as it is explained in Gadre’s paper here, his analysis of cusp excursions generalize to geodesic flows on complete non-compact finite-area hyperbolic surfaces and to Teichmüller geodesic flows on moduli spaces of flat surfaces.

As the reader can infer from Section 2 of Gadre’s paper, an important ingredient in his investigation of cusp excursions is the *exponential mixing* property for the corresponding geodesic flows.

Partly motivated by this fact, Vaibhav Gadre and I asked ourselves if the exponential mixing result for “Weil–Petersson like” geodesic flows on surfaces obtained by Burns–Masur–M.–Wilkinson could be explored to control cusp excursions of typical geodesics.

In this post, we record lower and upper bounds obtained together with V. Gadre on the depth of cusp excursions of typical “Weil–Petersson like” geodesics.

Remark 1As usual, all errors/mistakes are my sole responsibility.

Remark 2Our exposition follows closely Section 2 of Gadre’s paper.

**1. Ergodic averages of exponentially mixing flows**

Let be a flow on preserving a probability measure .

Suppose that has *exponential decay of correlations*, i.e., there are constants and such that

for all and all “smooth” real-valued observables in a Banach space containing all constant functions (e.g., is a Hölder or Sobolev space).

*Proof:* We write

By -invariance of , we get

The exponential decay of correlations (1) implies that

This proves the lemma.

**2. Effective ergodic theorem for fast mixing flows**

Suppose that is an exponentially mixing flow on (i.e., satisfies (1)).

Fix and denote .

Theorem 3Given , a function such that for each , and a sequence of non-negative functions, we have for -almost every that

for all sufficiently large (depending on ).

*Proof:* Given , let . Since , we get from Lemma 2 that

Therefore,

Consider the sequence and let . From the estimate (2) with and , and and , we get

and

By Borel–Cantelli lemma, the summability of the series for and the previous inequalities imply that for -almost every

and

for all sufficiently large (depending on ).

On the other hand, the non-negativity of the functions says that

for all . Hence,

for all .

It follows from this discussion that for -almost every and all sufficiently large (depending on )

whenever . Because as and for , given , the previous estimate says that for -almost every

for all sufficiently large (depending on and ). This proves the theorem.

**3. Bounds for certain cusp excursions**

Let be a compact surface with finitely many punctures equipped with a negatively curved Riemannian metric which is “asymptotically modelled” by surfaces of revolutions of profiles , , near the punctures: see this paper here for details.

In this setting, it was shown by Burns, Masur, M. and Wilkinson that the geodesic flow on is exponentially mixing with respect to the Liouville (volume) measure , i.e., for each , the estimate (1) holds for the space of -Hölder functions.

In this section, we want to explore the exponential mixing result of Burns–Masur–M.–Wilkinson to study the cusp excursions of .

For the sake of simplicity of exposition, we are going to assume that there is only one cusp where the metric is isometric to the surface of revolution of the profile for and .

Remark 3The general case can be deduced from the arguments below after replacing the geometric facts about the surfaces of revolution of (e.g., Clairaut’s relations, etc.) by their analogs in Burns–Masur–M.–Wilkinson article (e.g., quasi-Clairaut’s relation in Proposition 3.2, etc.).

Given a vector with base point near the cusp, let be the angle between and the direction pointing straight into the cusp of the surface of revolution of . Denote by the collar in around the cusp consisting of points whose -coordinate satisfies .

**3.1. Good initial positions for deep excursions**

Given a parameter , let . The next proposition says that any vector in generates a geodesic making a –*deep* excursion into the cusp in *bounded* time.

Proposition 4If , then the base point of has -coordinate for a certain time where depends only on .

*Proof:* By Clairaut’s relation, the -coordinate along satisfies

for all (during the cusp excursion).

Thus, the value of the -coordinate along is minimized when : at this instant . Since implies that and , the proof of the proposition will be complete once we can bound by a constant . As it turns out, this fact is not hard to establish from classical facts about geodesics on surfaces of revolution: see, for example, Equation (6) in Pollicott–Weiss paper.

**3.2. Smooth approximations of characteristic functions**

Take a smooth non-negative bump function equal to on and supported on such that . Similarly, take a smooth non-negative bump function equal to on and supported on such that .

The non-negative function is a smooth approximation of the characteristic function of :

- is supported on ;
- there exists a constant depending only on such that
- and
- .

**3.3. Deep cusp excursions of typical geodesics**

At this point, we are ready to use the effective ergodic theorem to show that typical geodesics perform deep cusp excursions:

Theorem 5For -almost every and for all sufficiently large (depending on and ), the base point of has -coordinate for a certain time . (Here, denotes any quantity slightly larger than .)

*Proof:* Fix , , . Let be a parameter to be chosen later and consider the function , for (where ).

The effective ergodic theorem (cf. Theorem 3) applied to the functions introduced in the previous subsection says that, for -almost every and all sufficiently large (depending on and ),

On the other hand, by construction, and for a certain constant and for all .

It follows that, for -almost every and all sufficiently large,

If , i.e., , the right-hand side of this inequality is strictly positive for all sufficiently large. Since the function is supported on , we deduce that if then, for -almost every and all sufficiently large, (where ) for some .

By plugging this information into Proposition 4, we conclude that, if

then, for -almost every and all sufficiently large, the -coordinate of is for some time (where is a constant).

This proves the desired theorem: indeed, we can take the parameter arbitrarily close to in the previous paragraph because as and .

**3.4. Very deep cusp excursions are atypical**

Closing this post, let us now show that an elementary argument *à la Borel–Cantelli* implies that a typical geodesic *doesn’t* perform very deep cusp excursions:

Theorem 6For -almost every and for all sufficiently large (depending on and ), the base point of has -coordinate for all times . (Here, denotes any quantity slightly smaller than .)

*Proof:* Let and be parameters to be chosen later, and denote .

By elementary geometrical considerations about surfaces of revolution (similar to the proof of Proposition 4), we see that if the base point of has -coordinate , then the base point of has -coordinate in for all .

Therefore, if we divide into intervals of sizes , then

Since the Liouville measure is -invariant and the surface of revolution of the profile has the property that the volume of the region is , we deduce that

for all . Because we need indices to cover the time interval , we obtain that

We want to study the set . We divide into and . Because , we just need to compute . For this sake, we observe that

and, *a fortiori*, thanks to (3). In particular,

Note that the series is summable when , i.e., In this context, Borel–Cantelli lemma implies that, for -almost every , the -coordinate is for all and all sufficiently large (depending on ). Since as , we conclude that if

then for -almost every , the -coordinate is for all and all sufficiently large (depending on ).

This ends the proof of the theorem: in fact, by letting , we can take arbitrarily close to in the previous paragraph.

Remark 4By Theorems 5 and 6, a typical geodesic enters the region while avoiding the region during the time interval (for all sufficiently large).Of course, the presence of a gap between and motivates the following question: is there anoptimalexponent such that a typical geodesic enters while avoiding during the time interval (for all sufficiently large)?

]]>

Remark 5Contrary to the cases discussed in Gadre’s paper, we can’t show that typically there is only one “maximal” cusp excursion in our current setting (of geodesic flows on negatively curved surfaces with cusps modelled by surfaces of revolution of profiles ). In fact, the exponential decay of correlations estimate proved by Burns–Masur–M.–Wilkinson is not strong enough to provide good quasi-independence estimates for consecutive cusp excursions (with the same quality of Lemma 2.11 in Gadre’s paper).

This article is part of the PhD thesis project of Rodolfo (under the supervision of Anton Zorich and myself), which started last September 2016. (In fact, one of my motivations to obtain a “Habilitation à Diriger des Recherches” degree last June 2, 2017 was precisely to be able to formally co-supervise Rodolfo’s PhD thesis project.)

In this (short) blog post, we discuss some aspects of Rodolfo’s solution to Zorich conjecture (and we refer to the preprint for the details).

**1. Statement of Zorich conjecture**

The study of Lyapunov exponents of the Kontsevich-Zorich cocycle (and, more generally, variations of Hodge structures) found many applications since the pioneer works of Zorich and Forni in the late nineties:

- Zorich and Forni described the deviations of ergodic averages of typical interval exchange maps and translation flows in terms of Lyapunov exponents;
- Avila and Forni used in 2007 the positivity of second Lyapunov of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures (among many other ingredients) to show that typical interval exchange transformations and translation flows are weak mixing;
- Delecroix, Hubert and Lelièvre confirmed in 2014 a conjecture of Hardy and Weber on the abnormal rate of diffusion of typical trajectories on -periodic Ehrenfest wind-tree models of Lorenz gases;
- Kappes and Möller completed in 2016 the classification of commensurability classes of non-arithmetic lattices of , , constructed by Deligne and Mostow in the eighties thanks to new invariants coming from Lyapunov exponents;
- etc.

The success of Zorich in describing such deviations of ergodic averages together with many numerical experiments led Kontsevich and him to conjecture that the Lyapunov exponents of the Kontsevich-Zorich (KZ) cocycle with respect to Masur-Veech measures are simple (i.e., their multiplicities are ).

Moreover, Zorich had in mind a specific way to establish the Kontsevich-Zorich conjecture: first, he conjectured that the so-called *Rauzy-Veech groups* (associated to a certain combinatorial description of the matrices of the KZ cocycle appearing along typical trajectories for the Masur-Veech measures) are Zariski-dense in the symplectic groups , ; then, he noticed that the works of Guivarc’h-Raugi and Goldsheid-Margulis on the simplicity of Lyapunov exponents for random products of matrices forming a Zariski-dense subgroup could be useful to deduce the “Kontsevich-Zorich simplicity conjecture” from his “Zariski density conjecture”.

After an important partial result of Forni in 2002, Avila and Viana famously established the Kontsevich-Zorich conjecture in 2007. Nevertheless, the arguments of Avila and Viana were slightly different from the scheme outline by Zorich: indeed, as they pointed out in Remark 6.12 of their paper, Avila and Viana avoided discussing the Zariski closure of Rauzy-Veech groups by showing that Rauzy-Veech groups are *pinching* and *twisting*, and that these two properties *suffice* to get the simplicity of the Lyapunov spectrum (i.e., Kontsevich-Zorich conjecture).

Remark 1It is worth to notice that Zariski density implies pinching and twisting, but the converse is not true in general.

In summary, the solution of the Kontsevich-Zorich conjecture by Avila and Viana via the pinching and twisting properties for Rauzy-Veech groups left open Zorich’s conjecture on the Zariski density of Rauzy-Veech groups.

Remark 2Besides giving stronger information about Rauzy-Veech groups (and, in particular, a new proof of Avila and Viana theorem), Zorich’s conjecture has other applications: for example, Magee recently showed that the validity of Zorich’s conjecture implies that the spectral gap / rate of mixing of the geodesic flow on congruence covers of connected components of the strata of moduli spaces of unit area translation surfaces is uniform.

**2. Hyperelliptic Rauzy-Veech groups**

As we already discussed in this blog, Avila, Yoccoz and myself were able to prove Zorich’s conjecture in the particular case of *hyperelliptic* Rauzy-Veech groups by showing the *stronger* statement that such groups contain an *explicit* finite-index subgroup of : roughly speaking, the Rauzy-Veech group is the subgroup of consisting of matrices whose reduction modulo two permute the basis vectors and .

As it turns out, the hyperelliptic Rauzy-Veech groups are associated to one of the three connected components of the so-called *minimal strata* (consisting of translation surfaces of genus with a unique conical singularity of total angle ): in fact, it was proved by Kontsevich and Zorich in 2003 that the minimal strata (of genus ) have three connected components called *hyperelliptic* , *even* spin and *odd* spin .

**3. Rauzy-Veech groups of minimal strata**

As a warm-up problem, we asked Rodolfo to perform numerical experiments with the Rauzy-Veech groups of the odd connected component of the minimal stratum in genus 3 and the even and odd connected components of the minimal stratum in genus 4. In particular, we told him to “compute” the indices of the reductions modulo two of such a Rauzy-Veech group in and .

After playing a bit with the matrices in his computer, Rodolfo announced (among many other things) that the index in of the Rauzy-Veech group of was 28.

This number ringed a bell because (as it is briefly explained here for instance) contains two orthogonal subgroups , resp. , of indices , resp. , consisting of matrices stabilizing a quadratic form with even, resp. odd Arf invariant of representing the reduction modulo two of the symplectic form. In particular, the fact that the number matches the index of suggest the conjecture that Rauzy-Veech groups of , resp. , is the pre-image of , resp. in under the reduction modulo two map .

Once we convinced ourselves about the plausibility of this conjecture, Rodolfo started working on the geometry of the corresponding *Rauzy diagrams* (graphs underlying the structure of the Rauzy-Veech groups) in order to figure out a systematic way of producing many particular matrices generating the desired candidate groups above.

As it turns out, Rodolfo did this in two steps (which occupy most [15 pages] of his preprint):

- first, he exploits the fact that the level two congruence subgroup of (i.e., the kernel of the natural map ) is generated by the squares of certain symplectic transvections to show that the Rauzy-Veech groups of the odd and even components of contain ; for this sake, he exhibits a rich set of loops in Rauzy diagrams inducing appropriate Dehn twists (giving “most” of the desired symplectic transvections).
- secondly, he proves that the reduction modulo two of the Rauzy-Veech groups of the odd, resp. even components of coincides with , resp. , by using the fact that and are generated by orthogonal transvections.

In summary, Rodolfo showed that the Rauzy-Veech groups of the odd and even components of are explicit subgroups of of indices and .

By putting this result together with the result by Avila, Yoccoz and myself for hyperelliptic Rauzy-Veech groups, we conclude that the Rauzy-Veech group of any connected component of a minimal stratum is a finite-index subgroup of and, *a fortiori*, a Zariski-dense subgroup of .

**4. Rauzy-Veech groups of general strata**

Philosophically speaking, a general translation surface of genus “differs” from a translation surface in the minimal stratum because of the relative homology produced by the presence of many conical singularities. In particular, if we “merge” conical singularities of , we should find a translation surface .

From the geometrical point of view, this philosophy was made rigorous by Kontsevich and Zorich in 2003: indeed, they formalized the notions of “merging” and “breaking” zeroes in order to “reduce” the classification of connected components of general strata to the case of connected components of minimal strata!

From the combinatorial point of view, Avila and Viana obtained the following combinatorial analog of Kontsevich-Zorich geometrical statement: we can merge conical singularities until we end up with a component of a minimal stratum in such a way that a “copy” of the Rauzy-Veech group of a component of a minimal stratum shows up inside any Rauzy-Veech group.

Hence, this result of Avila-Viana (or rather its variant stated as Lemma 6.3 in Rodolfo’s preprint) allows Rodolfo to conclude his proof of (a statement slightly stronger than) Zorich’s conjecture: the Rauzy-Veech group of any connected component of any stratum of the moduli space of unit area translation surfaces of genus is a finite-index subgroup of simply because the same is true for the connected components of the minimal stratum .

]]>By carefully analyzing Freiman’s argument, Cusick and Flahive constructed in 1989 a sequence converging to as such that for all , and, as it turns out, was the largest *known* element of .

In our recent preprint, Gugu and I described the structure of the complement of the Lagrange spectrum in the Markov spectrum near , and this led us to wonder if our description could be used to find new numbers in which are larger than .

As it turns out, Gugu and I succeeded in finding such numbers and we are currently working on the combinatorial arguments needed to extract the largest number given by our methods. (Of course, we plan to include a section on this matter in a forthcoming revised version of our preprint.)

In order to give a flavour on our construction of new numbers in , we will prove in this post that a certain number

belongs to .

**1. Preliminaries**

Let

be the usual continued fraction expansion.

We abbreviate periodic continued fractions by putting a bar over the period: for instance, . Moreover, we use subscripts to indicate the multiplicity of a digit in a sequence: for example, .

Given a bi-infinite sequence and , let

In this context, recall that the classical Lagrange and Markov spectra and are the sets

where

As we already mentioned, Freiman proved that

and Cusick and Flahive extended Freiman’s argument to show that the sequence

accumulating on has the property that for all . In particular,

was the largest known number in .

**2. A new number in **

In what follows, we will show that

Remark 1is a “good” variant of in the sense that it falls in a certain interval which can be proved to avoid the Lagrange spectrum: see Proposition 5 below.

Remark 2Note that , i.e., if we center our discussion at , then is almost 25 times bigger than .

Similarly to the arguments of Freiman and Cusick-Flahive, the proof of Theorem 1 starts by locating an appropriate interval centered at such that does not intersect the Lagrange spectrum.

In this direction, one needs the following three lemmas:

Lemma 2If contains any of the subsequences

- (a)
- (b)
- (c)
- (d)
- (e)
- (f)

then where indicates the position in asterisk.

*Proof:* See Lemma 2 in Chapter 3 of Cusick-Flahive’s book.

Lemma 3If contains any of the subsequences:

- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- (ix)
- (x)
- (xi)

then where indicates the position in asterisk.

*Proof:* See Lemma 1 in Chapter 3 of Cusick-Flahive’s book and also Lemma 3.2 of our preprint with Gugu.

Lemma 4If contains the subsequence:

- (xii)

then where indicates the position in asterisk.

*Proof:* In this situation,

thanks to the standard fact that if

and

with , then if and only if .

As it is explained in Chapter 3 of Cusick-Flahive’s book and also in the proof of Proposition 3.7 of our preprint with Gugu, Lemmas 2, 3 and 4 allow to show that:

does not intersect the Lagrange spectrum .

Of course, this proposition gives a natural strategy to exhibit new numbers in : it suffices to build elements of as close as possible to the right endpoint of .

Remark 3As the reader can guess from the statement of the previous proposition, the right endpoint of is intimately related to Lemma 4. In other terms, the natural limit of this method for producing the largest known numbers in is given by how far we can push to the right the boundary of .Here, Gugu and I are currently trying to optimize the choice of by exploiting the simple observation that the proof of Lemma 4 is certainly not sharp in our situation: indeed, we used the sequence

to bound , but we could do better by noticing that this sequence provides a pessimistic bound because contains a copy of the word (and, thus, , i.e., can’t coincide with on a large chunk when ).

Anyhow, Proposition 5 ensures that

does not belong to the Lagrange spectrum .

At this point, it remains only to check that belongs to the Markov spectrum.

For this sake, let us verify that

By items (a), (b), (c) and (e) of Lemma 2,

except possibly for with . Since

we have

for all . This proves that belongs to the Markov spectrum.

]]>In this article, we study the complement of the Lagrange spectrum in the Markov spectrum near a non-isolated point found by Freiman, and, as a by-product, we prove that its Hausdorff dimension is

Remark 1Currently, this paper deals exclusively with lower bounds on . In its next version, Gugu and I will include upper bounds on .

In what follows, we present a *streamlined* version of our proof of based on the construction of an *explicit* Cantor set with .

Remark 2W e refer to our paper for more refined informations about the structure of near .

**1. Perron’s characterization of the classical spectra **

Given a bi-infinite sequence and , let

Here,

is the usual continued fraction expansion, and

is the th convergent.

In 1921, Perron showed that the classical Lagrange and Markov spectra and are the sets

where

**2. Freiman’s number **

In 1973, Freiman showed that

In a similar vein, Theorem 4 in Chapter 3 of Cusick-Flahive book asserts that

for all . In particular, is not isolated in .

Remark 3As it turns out, is the largest known number in : see page 35 of Cusick-Flahive book.

In what follows, we shall revisit Freiman’s arguments as described in Chapter 3 of Cusick-Flahive book in order to prove the following result:

Theorem 1Consider the alphabet consisting of the words and . Then,

**3. A standard comparison tool **

In the sequel, we use the following standard comparison tool for continued fractions is the following lemma (cf. Lemmas 1 and 2 in Chapter 1 of Cusick-Flahive book):

- if and only if ;
- .

Remark 4For later use, note that Lemma 2 implies that if and for all , then when is odd, and when is even.

**4. Proof of Theorem 1
**

Similarly to the discussions in Cusick-Flahive book, we shall use the next lemma (extracted from Lemma 2 in Chapter 3 of this book):

Lemma 3If contains any of the subsequences

- (a)
- (b)
- (f)

then where indicates the position in asterisk.

*Proof:* If (a) occurs, then .

If (b) occurs, then Remark 4 implies that

If (f) occurs, then Remark 4 implies that

We shall also need the following fact:

Lemma 4If is a bi-infinite sequence such that

then .

*Proof:* See the proof of Theorem 4 in Chapter 3 of Cusick-Flahive book (especially the last paragraph at page 40).

These lemmas allow us to conclude the proof of Theorem 1 along the following lines.

Proposition 5Given a bi-infinite sequence

where for all and serves to indicate the zeroth position, then

*Proof:* On one hand, Remark 4 implies that

and

and items (a), (b) and (f) of Lemma 3 imply that

for all positions except possibly for with .

On the other hand,

so that for all . This proves the proposition.

At this point, the proof of Theorem 1 is complete: in fact, Proposition 5 and Lemma 4 together imply that

is contained in .

**5. Lower bounds on **

The Gauss map , (where is the fractional part of ) acts on continued fractions as a shift operator:

Therefore, we can use the iterates of the Gauss map to build a bi-Lipschitz map between the Cantor set introduced above and the dynamical Cantor set

Since the Hausdorff dimension is preserved by bi-Lipschitz maps, an immediate corollary of Theorem 1 is:

On the other hand, the Hausdorff dimension was estimated in Subsection 2.2 of this previous post here. In particular, it was shown that:

By putting Corollary 6 and Proposition 7, we conclude the desired estimate

in the title of this post.

]]>The wide range of applicability of dynamical Cantor sets is partly explained by the fact that several natural examples of Cantor sets are defined in terms of Dynamical Systems: for example, Cantor’s ternary set is

where is .

In some applications of dynamical Cantor sets, it is important to dispose of estimates on their Hausdorff dimensions: for instance, the celebrated work of Bourgain and Kontorovich on Zaremba’s conjecture needs particular types of dynamical Cantor sets with Hausdorff dimension close to one.

For this reason, a considerable literature on this topic was developed. Among the diverse settings covered by many authors, one finds the articles of Bumby, Hensley, …, Jenkinson-Pollicott, Falk-Nussbaum, where the so-called *thermodynamical methods* are exploited to produce approximations for the Hausdorff dimension of Cantor sets defined in terms of continued fraction expansions (i.e., Cantor sets of number-theoretical nature).

In general, the thermodynamical methods quoted above provide a sequence of fast-converging approximations for the Hausdorff dimension of dynamical Cantor sets: for instance, the algorithm described by Jenkinson-Pollicott here gives a sequence converging to at *super-exponential speed*, i.e., for some constants and , where is the Cantor set of real numbers whose continued fraction expansions contain only and .

In particular, the thermodynamical methods give good *heuristics* for the first several digits of the Hausdorff dimension of dynamical Cantor sets (e.g., if we list for and the first three digits of coincide for all , then it is likely that one has found the first three digits of ).

The heuristic bounds provided by the thermodynamical methods can be turned into rigorous estimates: indeed, one of the goals of the recent work of Jenkinson-Pollicott consists into rigorously computing the first 100 digits of .

However, the conversion of heuristic bounds into rigorous estimates is not always easy, and, for this reason, sometimes a *slowly* converging method producing two sequences and of *rigorous* bounds (i.e., for all ) might be interesting for practical purposes.

In this post, we explain a method described in pages 68 to 70 of Palis-Takens book giving explicit sequences converging slowly (e.g., for some constant and all ) towards , and, for the sake of comparison, we apply it to exhibit crude bounds on the Hausdorff dimensions of some Cantor sets defined in terms of continued fraction expansions.

**1. Dynamical Cantor sets of the real line **

Definition 1A -dynamical Cantor set is

where:

- is an expanding -map (i.e., for every ) from a finite union of pairwise disjoint closed intervals to the convex hull of ;
- is a Markov partition, that is, is the convex hull of the union of some of the intervals , and
- is topologically mixing, i.e., for some , for all .

Example 1As we already mentioned, Cantor’s ternary set is a dynamical Cantor set:

where is the affine map .

Example 2Let be a finite alphabet of finite words . The Cantor set

of real numbers whose continued fraction expansions are given by concatenations of the words in is a dynamical Cantor set.In fact, it is possible to construct intervals such that

where and is the Gauss map: see, e.g., this paper here for more explanations.

By definition, a dynamical Cantor set can be inductively constructed as follows. We start with the Markov partition given by the connected components of the domain of the expanding map defining . For each , we define as the collection of connected components of , .

For later use, for each , we denote by

** 1.1. Hausdorff dimension and box counting dimension **

Recall that the Hausdorff dimension and the box-counting dimension of a compact set are defined as follows.

The Hausdorff -measure of is

and the Hausdorff dimension of is

The box-counting dimension is

where is the smallest number of intervals of lengths needed to cover .

Exercise 1Show that .

** 1.2. Upper bound on the dimension of dynamical Cantor sets **

Let be a -dynamical Cantor set associated to an expanding map . Consider the quantities defined by

*Proof:* Let and fix . By definition, there exists such that

for all .

In other words, given , we can cover using a collection of intervals with such that every has length .

It follows from the definitions that, for each , the pre-images of the intervals under form a covering of by intervals of length . Therefore,

for all and , and, *a fortiori*,

for all and .

Hence, if we define , then

for all .

By iterating this argument times, we conclude that

for all .

Thus,

Since is arbitrary, we deduce from the previous inequality that

Because , we get that , and, *a fortiori*, .

** 1.3. Lower bound on the dimension of dynamical Cantor sets **

Let be a -dynamical Cantor set associated to an expanding map . Consider the sequence defined in the previous subsection and fix be a constant such that for all (e.g., certainly works).

Take such that for all and set

Remark 1If is a full Markov map, i.e., for all , then we can choose and .

Consider the quantities given by

*Proof:* Suppose by contradiction that and take .

By definition, , so that for every there is a finite cover of with

Note that any interval of length strictly smaller than

intersects at most one .

Thus, if we define , then each element of the cover of intersects at most one element of . Hence, if we define

then, given any , one has that has *fewer* elements than .

Consider such that for all . From the definitions, for each , we see that is a well-defined cover of such that

Since , we get that

We want to exploit this estimate to prove that

for some . In this direction, suppose that

In this case, the discussion above would imply

a contradiction because on one hand (by definition of ) and on the other hand (by our choice of ).

In summary, we assumed that , we considered an arbitrary cover of with

and we found a cover of with *fewer* elements than such that

By iterating this argument, we would end up with a cover of containing no elements, a contradiction. This proves that .

** 1.4. Slow convergence of towards the Hausdorff dimension **

Let be a -dynamical Cantor set associated to an expanding map . In general, the sequences discussed above converge slowly towards :

*Proof:* The so-called *bounded distortion property* (see Theorem 1 in Chapter 4 of Palis-Takens book or this blog post here) ensures the existence of a constant such that

for all and .

Let and, for all , define

In this setting, we have that

Therefore, , that is,

This proves the proposition.

**2. Hausdorff dimension of Gauss-Cantor sets **

In this section, we apply the previous discussion in the context of Gauss-Cantor sets , that is, the dynamical Cantor sets from Example 2 above.

For the sake of simplicity of exposition, we will not describe the calculation of the sequences and approaching for a general , but we shall focus on two particular examples.

** 2.1. Some bounds on the Hausdorff dimension of **

Consider the alphabet consisting of the words and . The corresponding Cantor set is the Cantor set — denoted by in the beginning of this post — of real numbers whose continued fraction expansions contain only and .

The theory of continued fractions (see Moreira’s paper and Cusick-Flahive book) says that is the dynamical Cantor set associated to the restriction of the Gauss map to , where

Note that the functions and defined on the interval are the inverse branches of .

Remark 2Here, denotes the infinite word obtained by periodic repetition of the block .

By applying to the Markov partition , we deduce that consists of the intervals , , with extremities

The quantities and are not hard to compute using the following remarks. First, is monotone on each (because are Möbius transformations induced by integral matrices with determinant ), so that the values and are attained at the extremities of . Secondly, the derivative of the Gauss map is . By combining these facts, we get that

and

Hence, the sequences and defined in the previous section are the solutions of the equations

and

(Here, we used Remark 1 and the fact that is a full Markov map in order to get the equation of .)

Of course, these equations allow to find the first few terms of the sequences and approaching with some computer-aid: for example, this Mathematica routine here shows that

and

In particular,

Remark 3The approximations and of (obtained from computing with the words in ) are very poor in comparison with the approximation

provided by the thermodynamical method of Jenkinson-Pollicott (also based on calculations with the words in ). In fact, Jenkinson-Pollicott showed in this paper here that the first 100 digits of are

Thus, the first 18 digits of accurately describe (while only the first two digits of and the first digit of are accurate).

** 2.2. Some bounds on the Hausdorff dimension of **

Let us consider now the alphabet consisting of the words and . The theory of continued fractions says that the convex hull of is the interval with extremities and . The images and of under the inverse branches

of the first two iterates of the Gauss map provide the first step of the construction of the Cantor set . In general, given , the collection of intervals of the th step of the construction of is given by

Hence, the interval associated to a string has extremities and .

Similarly as in the previous subsection, we conclude that the quantities and are given by

and

Thus, and are the solutions of

and

Once more, we can calculate the first terms of the sequences and with some computer-aid: this Mathematica routine here reveals that

and

In particular,

Remark 4After implementing of the Jenkinson-Pollicott method with this Mathematica routine here, we get the following approximations for :

The super-exponential convergence of Jenkinson-Pollicott method suggests that

Note that the first two digits of this approximation are accurate (because the first two digits of and coincide).

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