In a series of two posts, we will revisit our previous discussion on the exponential mixing property for hyperbolic flows via a technique called Dolgopyat’s estimate.

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 {T_t:\Delta_r\rightarrow \Delta_r}, {t\in\mathbb{R}_+}, associated to a base dynamics (discrete-time dynamical system) {T:\Delta\rightarrow\Delta} and a roof function {r:\Delta\rightarrow\mathbb{R}^+} in the following way. We consider {\Delta_r:=(\Delta\times\mathbb{R}^+)/\sim} where {\sim} is the equivalence relation induced by {(T(x),0)\sim (x,r(x))}, and we let {T_t} be the semiflow on {\Delta_r} induced by

\displaystyle (x,s)\in \Delta\times\mathbb{R}^+\mapsto (x,s+t)\in\Delta\times\mathbb{R}^+

Geometrically, {T_t}, {0\leq t<\infty} flows up the point {(x,s)}, {0\leq s<r(x)}, linearly (by translation) in the fiber {\{x\}\times\mathbb{R}^+} until it hits the “roof” (the graph of {r}) at the point {(x,r(x))}. At this moment, one is sent back (by the equivalence relation {\sim}) to the basis {\Delta\times\{0\}} at the point {(T(x),0)\sim (x,r(x))}, and the semiflow restarts again.

A more concise way of writing down {T_t} is the following: denoting by {\Delta_r:=\{(x,t):x\in \Delta, 0\leq t<r(x)\}}, one defines {T_t(x,s) := (T^n x, s+t-r^{(n)}(x))} where {r^{(n)}(x)} is the Birkhoff sum

\displaystyle r^{(n)}(x):=\sum\limits_{k=0}^{n-1} r(T^k x) \ \ \ \ \ (1)

and {n} is the unique integer such that

\displaystyle r^{(n)}(x)\leq s+t<r^{(n+1)}(x)

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

Remark 1 Avila-Gouëzel-Yoccoz work in greater generality than the setting of this post: in fact, they allow {\Delta} to be a John domain and they prove results for excellent hyperbolic semiflows (which are more common in “nature”), but we will always take {\Delta=(0,1)} 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 1 Let {\Delta=(0,1)}, {Leb} be the Lebesgue measure on {\Delta}, and {\{\Delta^{(l)}\}_{l\in L}} be a finite or countable partition of {\Delta} modulo zero into open subintervals. We say that {T:\bigcup\limits_{l\in L} \Delta^{(l)}\rightarrow \Delta} is an uniformly expanding Markov map if

  • {\{\Delta^{(l)}\}} is a Markov partition: for each {l\in L}, the restriction of {T} to {\Delta^{(l)}} is a {C^1}-diffeomorphism between {\Delta^{(l)}} and {\Delta};
  • {T} is expanding: there exist a constant {\kappa>1} and, for each {l\in L}, a constant {C(l)>1} such that {\kappa\leq|T'(x)|\leq C(l)} for each {x\in\Delta^{(l)}};
  • {T} has bounded distortion: denoting by {J(x)=1/|T'(x)|} the inverse of the Jacobian of {T} and by {\mathcal{H}=\{(T|_{\Delta^{(l)}})^{-1}\}_{l\in L}} the set of inverse branches of {T}, we require that {\log J} is a {C^1} function on each {\Delta^{(l)}} and there exists a constant {C>0} such that

    \displaystyle \left|\frac{h''(x)}{h'(x)}\right| = |D((\log J)\circ h)(x)|\leq C

    for all {h\in \mathcal{H}} and {x\in \Delta}. (This condition is also called Renyi condition in the literature.)

Remark 2 Araújo 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 {\log J} is a Hölder function such that the Hölder constant of {\log J\circ h} is uniformly bounded for all {h\in \mathcal{H}}.

Example 1 Let {\Delta=\Delta^{(0)}\cup\Delta^{(1)}=(0,1/2)\cup(1/2,1)} be the finite partition (mod. {0}) of {\Delta} provided by the two subintervals {\Delta^{(l)}=(\frac{l}{2}, \frac{l+1}{2})}, {l=0, 1}. The map {T:\Delta^{(0)}\cup\Delta^{(1)}\rightarrow\Delta} given by {T(x)=2x-l} for {x\in\Delta^{(l)}} is an uniformly expanding Markov map (preserving the Lebesgue measure {Leb}).

An uniformly expanding map {T} preserves an unique probability measure {\mu} which is absolutely continuous with respect to the Lebesgue measure {Leb}. Moreover, the density {d\mu/dLeb} is a {C^1} function whose values are bounded away from {0} and {\infty}, and {\mu} 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

\displaystyle Lu(x) = \sum\limits_{T(y)=x} J(y) u(y) = \sum\limits_{h\in\mathcal{H}} J(hx) u(hx)

(as it was discussed in Theorem 1 of Section 1 of this post in the case of a finite Markov partition {\{\Delta^{(l)}\}_{l\in L}}, {\# L<\infty})

Definition 2 Let {T:\bigcup\limits_{l\in L} \Delta^{(l)}\rightarrow \Delta} be an uniformly expanding Markov map. A function {r:\bigcup\limits_{l\in L}\Delta^{(l)}\rightarrow\mathbb{R}^+} is a good roof function if

  • there exists a constant {\varepsilon>0} such that {r(x)\geq\varepsilon} for all {x};
  • there exists a constant {C>0} such that {|D(r\circ h)(x)|\leq C} for all {x} and all inverse branch {h\in\mathcal{H}} of {T};
  • {r} is not a {C^1}coboundary: it is not possible to write {r = \psi + \phi\circ T - \phi} where {\psi:\Delta\rightarrow\mathbb{R}} is constant on each {\Delta^{(l)}} and {\phi:\Delta\rightarrow\mathbb{R}} is {C^1}.

Remark 3 Intuitively, the condition that {r} 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 3 A good roof function {r:\bigcup\limits_{l\in L}\Delta^{(l)}\rightarrow\mathbb{R}^+} has exponential tails if there exists {\sigma_0 > 0} such that {\int_{\Delta} e^{\sigma_0 r} d Leb < \infty}.

The suspension flow {T_t} associated to an uniformly expanding Markov map {T:\bigcup\limits_{l\in L}\Delta^{(l)}\rightarrow\Delta} and a good roof function {r:\bigcup\limits_{l\in L}\Delta^{(l)}\rightarrow\mathbb{R}^+} with exponential tails preserves the probability measure

\displaystyle \mu_r:=\mu\otimes Leb/\mu\otimes Leb(\Delta_r)

on {\Delta_r}. Note that {\mu_r} is absolutely continuous with respect to {Leb_r:= Leb\otimes Leb} (because {\mu} is absolutely continuous with respect to {Leb}).

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

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

2. Statement of the exponential mixing result

Let {(T_t)_{t\in\mathbb{R}}} be an expanding semiflow.

Theorem 4 There exist constants {C>0}, {\delta>0} such that

\displaystyle \left|\int U\cdot V\circ T_t \, d Leb_r - \left(\int U \, d Leb_r\right) \left(\int V\,d\mu_r\right)\right|\leq C e^{-\delta t}\|U\|_{C^1}\|V\|_{C^1}

for all {t\geq 0} and for all {U, V\in C^1(\Delta_r)}.

Remark 6 By applying this theorem with {U(x,t)\cdot \frac{d\mu}{d Leb}(x)} in the place of {U}, we obtain the classical exponential mixing statement:

\displaystyle \left|\int U\cdot V\circ T_t \, d\mu_r - \left(\int U \, d \mu_r\right) \left(\int V\,d\mu_r\right)\right|\leq C e^{-\delta t}\|U\|_{C^1}\|V\|_{C^1}

Remark 7 This theorem is exactly Theorem 7.3 in AGY paper except that they work with observables {U} and {V} belonging to Banach spaces {\mathcal{B}_0} and {\mathcal{B}_1} which are slightly more general than {C^1} (in the sense that {C^1\subset \mathcal{B}_0\subset \mathcal{B}_1}). 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.

Read More…

Some of the partial advances obtained by Jacob Palis, Jean-Christophe Yoccoz and myself on the computation of Hausdorff dimensions of stable and unstable sets of non-uniformly hyperbolic horseshoes (announced in this blog post here and this survey article here) are based on the following lemma:

Lemma 1 Let {f:B\rightarrow\mathbb{R}^2} be a {C^1} diffeomorphism from the closed unit ball {B:= \{(x,y)\in\mathbb{R}^2: x^2 + y^2\leq 1\}} of {\mathbb{R}^2} into its image.Let {K\geq 1} and {L\geq 1} be two constants such that {\|Df(p)\|\leq K} and {\textrm{Jac}(f)(p):=|\det Df(p)|\leq L} for all {p\in B}.

Then, for each {1\leq d\leq 2}, the {d}-dimensional Hausdorff measure {H^d_{\sqrt{2}}(f(B))} at scale {\sqrt{2}} of {f(B)} satisfies

\displaystyle H^d_{\sqrt{2}}(f(B)) := \inf\limits_{\substack{\bigcup\limits_{i\in \mathbb{N}} U_i \supset f(B), \\ \textrm{diam}(U_i)\leq \sqrt{2}}}\sum\limits_{i\in\mathbb{N}}\textrm{diam}(U_i)^d \leq 170\pi \cdot \max\{K,L\}^{2-d} \cdot L^{d-1} \ \ \ \ \ (1)

Remark 1 In 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

\displaystyle H^d_{r\sqrt{2}}(g(B_r))\leq 170\pi\cdot r^d\cdot K^{2-d}\cdot L^{d-1}

where {B_r} is the ball of radius {r} centered at the origin and {g} is a {C^1} diffeomorphism such that {\|Dg\|\leq K} and {\textrm{Jac}(g)\leq L} for {1\leq L\leq K}. Of course, this estimate is deduced from the lemma above by scaling, i.e., by applying the lemma to {f = h_r^{-1}\circ g\circ h_r } where {h_r:\mathbb{R}^2\rightarrow\mathbb{R}^2} is the scaling {h_r(p)=rp}.

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 {170\pi}]) happens when {f} is an affine hyperbolic conservative map {f_K(x,y)=(Kx,\frac{1}{K}y)} (say {K\geq 1}): indeed, since {[-\sqrt{\frac{1}{2}}, \sqrt{\frac{1}{2}}]\times [-\sqrt{\frac{1}{2}}, \sqrt{\frac{1}{2}}]\subset B\subset [-1,1]\times [-1,1]}, the most “economical” way to cover {f_K(B)} using a countable collection of sets of diameters {\leq \sqrt{2}} is basically to use {K^2} squares of sizes {1/K} (which gives an estimate {H^d_{\sqrt{2}}(f(B))\leq K^2(1/K)^d = K^{2-d}}).

However, in the context of (expectional subsets of stable sets of) non-uniformly hyperbolic horseshoes, we deal with maps {f} 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 {f} 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 {f(B)} with dyadic squares on {\mathbb{R}^2}, we can interpret the measure {H^d_{\sqrt{2}}(f(B))} as a sort of {L^d}-norm of a certain function. Since {1\leq d\leq 2}, we can control this {L^d}-norm in terms of the {L^1} and {L^2} norms (by interpolation). As it turns out, the {L^1}-norm, resp. {L^2}-norm, is controlled by the features of the derivative {Df}, resp. Jacobian determinant {Jac(f)}, and this morally explains the estimate (1).

Let us now turn to the details of this argument. Denote by {U:=f(B)} and {\partial U} its boundary. For each integer {k\geq 0}, let {\Delta_k} be the collection of dyadic squares of level {k}, i.e., {\Delta_k} is the collection of squares of sizes {1/2^k} with corners on the lattice {(1/2^k)\cdot\mathbb{Z}^2}.

Consider the following recursively defined cover of {U}. First, let {\mathcal{C}_0} be the subset of squares {Q\in \Delta_0} such that

\displaystyle \textrm{area}(Q\cap U)\geq \frac{1}{5}

Next, for each {k>0}, we define inductively {\mathcal{C}_k} as the subset of squares {Q\in\Delta_k} such that {Q} is not contained in some {Q'\in\mathcal{C}_l} for {0\leq l < k}, and {Q} intersects a significant portion of {U} in the sense that

\displaystyle \textrm{area}(Q\cap U)\geq \frac{1}{5}\textrm{area}(Q) \ \ \ \ \ (2)

In other words, we start with {U} and we look at the collection {\mathcal{C}_0} of dyadic squares of level {0} intersecting it in a significant portion. If the squares in {\mathcal{C}_0} suffice to cover {U}, we stop the process. Otherwise, we consider the dyadic squares of level {0} not belonging to {\mathcal{C}_0}, we divide each of them into four dyadic squares of level {1}, and we build a collection {\mathcal{C}_1} of such dyadic squares of level {1} intersecting in a significant way the remaining part of {U} not covered by {\mathcal{C}_0}, etc.

Remark 2 In this construction, we are implicitly assuming that {U=f(B)} is not entirely contained in a dyadic square {Q\in\bigcup\limits_{k=0}^{\infty}\Delta_k}. In fact, if {U\subset Q}, then the trivial bound {H^d_{\sqrt{2}}(U)\leq \textrm{diam}(Q)^d\leq (\sqrt{2})^d\leq 2} (for {1\leq d\leq 2}) is enough to complete the proof of the lemma.

In this way, we obtain a countable collection {\bigcup\limits_{k=0}^{\infty} \mathcal{C}_k:=(U_i)_{i\in\mathbb{N}}} covering {U=f(B)} such that {\textrm{diam}(U_i)\leq \sqrt{2}} and

\displaystyle H^d_{\sqrt{2}}(f(B))\leq \sum\limits_{i}\textrm{diam}(U_i)^d = \sum\limits_{k=0}^{\infty} N_k\left(\frac{1}{2^k}\right)^d \ \ \ \ \ (3)

where {N_k:=(\sqrt{2})^d\#\mathcal{C}_k}.

By thinking of this expression as a {L^d}-norm and by applying interpolation between the {L^1} and {L^2} norms, we obtain that

\displaystyle \sum\limits_{k=0}^{\infty} N_k\left(\frac{1}{2^k}\right)^d\leq \left(\sum\limits_{k=0}^{\infty} \frac{N_k}{2^k}\right)^{2-d} \left(\sum\limits_{k=0}^{\infty} \frac{N_k}{(2^k)^2}\right)^{d-1} \ \ \ \ \ (4)

This reduces our task to estimate these {L^1} and {L^2} norms. We begin by observing that the {L^2}-norm is easily controlled in terms of the Jacobian of {f} (thanks to the condition (2)):

\displaystyle \sum\limits_{k=0}^{\infty} \frac{N_k}{(2^k)^2} = (\sqrt{2})^d\sum\limits_{k}\sum\limits_{Q\in\mathcal{C}_k} \textrm{area}(Q) \ \ \ \ \ (5)

\displaystyle \begin{array}{rcl} &\leq & (\sqrt{2})^d\sum\limits_{k}\sum\limits_{Q\in\mathcal{C}_k} 5\cdot \textrm{area}(Q\cap U) \\ &\leq& 10 \cdot \textrm{area}(U) = 10 \int_B \textrm{Jac}(f) \\ &\leq& 10\pi\cdot L \end{array}

for any {1\leq d\leq 2}. In particular, we have that

\displaystyle N_0\leq 10\pi L

From this estimate, we see that the {L^1}-norm satisfies

\displaystyle \sum\limits_{k=0}^{\infty} \frac{N_k}{2^k} = N_0+\sum\limits_{k=1}^{\infty} \frac{N_k}{2^k}\leq 10\pi L+\sum\limits_{k>0} \frac{N_k}{2^k} \ \ \ \ \ (6)

Thus, we have just to estimate the series {\sum\limits_{k>0} \frac{N_k}{2^k}}. We affirm that this series is controlled by the derivative of {f}. In order to prove this, we need the following claim:

Claim. For each {k>0} and {Q\in\mathcal{C}_k}, one has

\displaystyle \textrm{length}(Q\cap \partial U)\geq \frac{1}{20}\cdot\frac{1}{2^k} \ \ \ \ \ (7)

Proof of Claim. Note that {U} can not contain {Q}: indeed, since {Q\subset Q'} for some dyadic square {Q'\in\Delta_{k-1}} of level {k-1\geq 0} (and, thus, {4\cdot \textrm{area}(Q) = \textrm{area}(Q')}), if {Q\subset U}, then {\textrm{area}(Q'\cap U)\geq \textrm{area}(Q\cap U) = \textrm{area}(Q)=\frac{1}{4}\textrm{area}(Q')}, a contradiction with the definition of {Q\in\mathcal{C}_k}. Because we are assuming that {U} is not contained in {Q} (cf. Remark 2) and we also have that {Q} intersects (a significant portion of) {U}, we get that

\displaystyle \partial U\cap \partial Q\neq \emptyset

For the sake of contradiction, suppose that {\textrm{length}(\partial U\cap Q)<\frac{1}{20\cdot 2^k}}. Since {\partial U} intersects {\partial Q}, the {\frac{1}{20\cdot 2^k}}-neighborhood {V_k} of {\partial Q} contains {\partial U\cap Q}. This means that

  • (a) either {Q-V_k} is contained in {U}
  • (b) or {Q-V_k} is disjoint from {U}

However, we obtain a contradiction in both cases. Indeed, in case (a), we get that a dyadic square {Q'} of level {k-1} containing {Q} satsifies

\displaystyle \textrm{area}(Q'\cap U)\geq \textrm{area}(Q-V_k) = \left(1-2\cdot\frac{1}{20}\right)^2\textrm{area}(Q) = \frac{81}{400}\textrm{area}(Q'),

a contradiction with the definition of {Q\in\mathcal{C}_k}. Similarly, in case (b), we obtain that

\displaystyle \textrm{area}(Q\cap U)\leq \textrm{area}(Q\cap V_k) = \left(1-\frac{81}{100}\right)\textrm{area}(Q) < \frac{1}{5}\textrm{area}(Q),

a contradiction with (2).

This completes the proof of the claim. {\square}

Coming back to the calculation of the series {\sum\limits_{k>0} N_k/2^k}, we observe that the estimate (7) from the claim and the fact that {\|Df\|\leq K} imply:

\displaystyle \begin{array}{rcl} \sum\limits_{k>0} \frac{N_k}{2^k} &=& (\sqrt{2})^d \sum\limits_{k>0}\sum\limits_{Q\in\mathcal{C}_k} \frac{1}{2^k} \\ &\leq& 20(\sqrt{2})^d\sum\limits_{k>0}\sum\limits_{Q\in\mathcal{C}_k}\textrm{length}(\partial U\cap Q) \\ &\leq& 20(\sqrt{2})^d 2 \cdot \textrm{length}(\partial U) \\ &\leq& 80 K\cdot \textrm{length}(\partial B) = 160\pi K \end{array}

By plugging this estimate into (6), we deduce that the {L^1}-norm verifies

\displaystyle \sum\limits_{k=0}^{\infty} \frac{N_k}{2^k}\leq 170\pi \max\{K,L\} \ \ \ \ \ (8)

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

\displaystyle H^d_{\sqrt{2}}(f(B))\leq (170\pi)^{2-d}(10\pi)^{d-1}\max\{K,L\}^{2-d} L^{d-1}\leq 170\pi \max\{K,L\}^{2-d} L^{d-1}

This ends the proof of the lemma.

Posted by: matheuscmss | April 24, 2015

Some comments on the conjectures of Ivanov and Putman-Wieland

During the graduate workshop on moduli of curves (organized by Samuel Grushevsky, Robert Lazarsfeld, and Eduard Looijenga last July 2014), Alex Wright gave a minicourse on the {SL(2,\mathbb{R})}-orbits on moduli spaces of translation surfaces (the videos of the lectures and the corresponding lecture notes are available here and here).

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

Let {S} be a ramified finite cover of the two-torus {T} (say branched at only one point {0\in T}). Denote by {H} the subspace of {H_1(S,\mathbb{Q})} generated by the homology classes of all simple closed loops on {S} covering such a curve on {T}.

Question 1. Is it true that one always has {H=H_1(S,\mathbb{Q})} 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 {g\geq 3} and {n\geq0}. Consider {\Gamma} a finite-index subgroup of the mapping-class group {\textrm{Mod}_{g,n}} of isotopy classes of homeomorphisms of a genus {g} surface {S} fixing pointwise a set {\Sigma=\{x_1,\dots, x_n\}} of marked points. Then, there is no surjective homomorphism from {\Gamma} to {\mathbb{Z}}.

Remark 1 This 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 {\mathbb{Z}} because they satisfy Kazhdan property (T). Nevertheless, Jorgen Andersen recently proved that the mapping-class groups {\textrm{Mod}_{g,n}} do not have Kazhdan property (T) when {g\geq 2}.

Remark 2 It was proved by John McCarthy and Feraydoun Taherkhani that the analog for {g=2} 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 {g\geq 2} and {n\geq 0}. Given a finite-index characteristic subgroup {K} of the fundamental group {\pi_1(S_{g,n}, x_{n+1})} of a surface {S=S_{g,n}} of genus {g} with {n} punctures {x_1, \dots, x_n}, denote by {S_K\rightarrow S} the associated finite cover, and let {\overline{S_K}} be the compact surface obtained from {S_K} by filling its punctures.

Then, the natural action on {H_1(\overline{S_K},\mathbb{Q})-\{0\}} of the group of lifts to {\overline{S_K}} of isotopy classes of diffeomorphisms of {S_{g,n}} fixing {x_1,\dots, x_{n+1}} pointwise has no finite orbits.

Remark 3 This 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 {S\rightarrow T}.

Remark 4 The analog of Putman-Wieland conjecture in genus {g=1} 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 5 In the context of Putman-Wieland conjecture, one has a representation {\textrm{Mod}_{g,n+1}\rightarrow \textrm{Aut}(H_1(\overline{S_K}, \mathbb{Q}))} (induced by the lifts of elements of {\textrm{Mod}_{g,n+1}} to {\overline{S_K}}). This representation is called a higher Prym representation by Putman-Wieland. In this language, Putman-Wieland conjecture asserts that higher Prym representations have no non-trivial finite orbits when {g\geq 2} and {n\geq 0}.

Secondly, they proved that:

Theorem 1 (Putman-Wieland) Fix {g\geq 3} and {n\geq 0}.

  • (a) If Putman-Wieland conjecture holds for every finite-index characteristic subgroup {K} of {\pi_1(S_{g-1, n+1}, x_{n+2})}, then Ivanov conjecture is true for any finite-index sugroup {\Gamma} of {\textrm{Mod}_{g,n}}.
  • (b) If Ivanov conjecture holds for every finite-index subgroup of {\textrm{Mod}_{g,n+2}}, then Putman-Wieland conjecture is true for any finite-index characteristic subgroup {K} of {\pi_1(S_{g,n+1}, x_{n+2})}.

Moreover, if Ivanov conjecture is true for all finite-index subgroups of {\textrm{Mod}_{g,n}} for all {n\geq0}, then it is also true for all finite-index subgroups of {\textrm{Mod}_{G,m}} with {G\geq g}, {m\geq 0}.

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 {g\geq 3} and {n\geq 0}, then it suffices to prove Putman-Wieland conjecture for {g=2} (and all {n\geq 0}); indeed, this is so because item (a) of Putman-Wieland theorem would imply that Ivanov conjecture is true for {g=3} (and all {n\geq0}) 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 {g\geq 3} and {m\geq 2}, then it suffices to construct a counterexample to Putman-Wieland conjecture for {g=3} and {n=m-1}.

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 {g=2} (and {n=6}).

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

Remark 6 This shows that Putman-Wieland’s strategy (I) above does not work (because their conjecture is false in genus {2}). Of course, this does not mean that Ivanov conjecture is false: in fact, by Putman-Wieland strategy (II), one needs a counterexample to Putman-Wieland conjecture in genus {g\geq 3} (rather than in genus {g=2}). Here, it is worth to point out that Artur, Jean-Christophe and I have no good candidates of counterexamples to Putman-Wieland conjecture in genus {g\geq 3} and/or Ivanov conjecture.

Below the fold, we focus on the case {g=2} and {n=6} of Putman-Wieland conjecture.

Update (September 7, 2015): Last June 2015, Eduard gave a talk on algebro-geometrical aspects of mapping-class groups, and he wrote the following summary here where the connections between (a natural generalization of) Question 1 and the conjectures of Ivanov and Putman-Wieland are discussed.

Read More…

Last March 25th, Sébastien Gouëzel gave the talk “Subadditive cocycles and horofunctions” at the Ergodic Theory and Dynamical Systems seminar of LAGA , Université Paris 13.

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 {\phi:\mathbb{R}^d\rightarrow\mathbb{R}^d} be a weak contraction of the Euclidean space {(\mathbb{R}^d, \|.\|)} in the sense that

\displaystyle \|\phi(x)-\phi(y)\|\leq \|x-y\|

for all {x,y\in\mathbb{R}^d}.Then, the sequence

\displaystyle \frac{\phi^n(0)}{n}

converges as {n\rightarrow\infty}.

Remark 1 The origin {0\in\mathbb{R}^d} can be replaced by any point {x_0\in\mathbb{R}^n} because

\displaystyle \|\phi^n(x_0)-\phi^n(0)\|\leq \|x_0\|

so that

\displaystyle \lim\limits_{n\rightarrow\infty}\frac{\phi^n(x_0) - \phi^n(0)}{n} = 0

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).

Read More…

Posted by: matheuscmss | March 15, 2015

First Bourbaki seminar of 2015 (III): Ambrosio’s talk

For the third (and last) installment of this series of posts (which started here) on the first Bourbaki seminar of 2015, we will discuss a talk that brought me back some good memories from the time I did my PhD at IMPA when I took a course on Geometric Measure Theory (taught by Hermano Frid) based on the books of Evans-Gariepy (for the introductory part of the course) and Guisti (for the core part of the course).

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 {m} and codimension {n} is the following. Given a {(m+n)}-dimensional Riemannian manifold and a {(m-1)}-dimensional compact embedded oriented submanifold {\Gamma\subset M} (without boundary), find a {m}-dimensional embedded oriented submanifold {\Sigma\subset M} with boundary {\partial\Sigma=\Gamma} such that

\displaystyle \textrm{vol}_m(\Sigma)\leq \textrm{vol}_m(\widetilde{\Sigma})

for all oriented {m}-dimensional submanifold {\widetilde{\Sigma}} with {\partial\widetilde{\Sigma} = \Gamma}. (Here, {\textrm{vol}_m(A)} denotes the {m}-dimensional volume of {A}).

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 {m} and codimension {n}).

Example 1 The 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 {m=2} and the boundary {\Gamma} is circular (i.e., {\Gamma} is parametrized by the round circle {S^1=\{ (x,y)\in\mathbb{R}^2: x^2+y^2=1 \}}). Unfortunately, the techniques employed by Douglas and Radó do not work for arbitrary dimension {m} and codimension {n}.

Example 2 The following example gives an idea on the difficulties that one might found while trying to solve Plateau’s problem (in the formulation given above).Let us consider the case {m=n=2}, {\Gamma\subset\mathbb{R}^4\simeq \mathbb{C}^2} defined as

\displaystyle \Gamma:=\{(\zeta^2,\zeta^3)\in\mathbb{C}^2: |\zeta|=1\}

The singular immersed disk

\displaystyle D:=\{(z, w)\in\mathbb{C}^2: z^3=w^2, |z|\leq 1\}

satisfies {\partial D=\Gamma} and the so-called calibration method can be applied to prove that

\displaystyle \mathcal{H}^2(D)<\mathcal{H}^2(\Sigma)

for all smooth oriented {2}-dimensional submanifold {\Sigma\subset\mathbb{R}^4} with {\partial\Sigma = \Gamma}. (Here, {\mathcal{H}^2} stands for the {2}-dimensional Hausdorff measure on {\mathbb{R}^4}.)

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 {n=1}, and Federer and Fleming introduced the notion of currents.

Remark 1 The “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 ({m}-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 {2} at least:

Theorem 1 (Almgren) Let {T} be an integral rectifiable {m}-dimensional current in a {(m+n)}-dimensional {C^5} Riemannian manifold {M^{m+n}}.If {T} is area-minimizing (i.e., a weak solution to Plateau’s problem), then there exists a closed subset {\textrm{sing}(T)} of {T} such that:

  • {\textrm{sing}(T)} has codimension {\geq 2}: the Hausdorff dimension of {\textrm{sing}(T)} is {\leq (m-2)}, and
  • {\textrm{sing}(T)} is the singular set of {T}: the subset {T\cap (M-(\textrm{supp}(\partial T)\cup\textrm{sing}(T))} is induced by a smooth oriented {m}-dimensional submanifold of {M}.

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 {1/3}) 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 {n=1}; 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 {n\geq 2}; in particular, we will sketch the proof of Theorem 1 above.

Remark 2 For 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., {M^{m+n} = \mathbb{R}^{m+n}}. In fact, this is not a great loss in generality because the statements and proofs in the Euclidean setting {M^{m+n} = \mathbb{R}^{m+n}} can be adapted to arbitrary Riemannian manifolds {M^{m+n}} with almost no extra effort.

Read More…

Posted by: matheuscmss | March 7, 2015

Journée Surfaces Plates

Luca Marchese and I are organizing a one-day event called “Journée Surfaces Plates” at LAGA, Université Paris 13 next May 20th, 2015.

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”.

Posted by: matheuscmss | February 28, 2015

First Bourbaki seminar of 2015 (II): Carron’s talk

For the second installment of this series of posts (which started here) on the first Bourbaki seminar of 2015, we will discuss Gilles Carron talk entitled “New utilisation of the maximum principles in Geometry (after B. Andrews, S. Brendle, J. Clutterbuck)”. Here, besides the original works of Andrews-Clutterbuck and Brendle (quoted below), the main references are the video of Carron’s talk and his lecture notes (both in French).

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

1. Introduction

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

  • a first order information: the gradient of {f} at {p} vanishes; and
  • a second order information: the Hessian of {f} at {p} 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 {f}, one investigates the local maxima of a “well-chosen” function {g} on the double of variables (e.g., {g:M\times M\rightarrow\mathbb{R}}). In this way, we have new constraints because the gradient and Hessian of {g} depend on more variables than those of {f}.

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) Let {\Omega\subset\mathbb{R}^n} be a convex domain of diameter {D}. Consider the Schrödinger operator {-\Delta+V} where {\Delta=\sum\limits_{i=1}^n\frac{\partial^2}{\partial^2 x_i}} is the Laplacian operator and {V} is the operator induced by the multiplication by a convex function {V:\Omega\rightarrow\mathbb{R}}.Recall that the spectrum of {-\Delta+V} with respect to Dirichlet condition on the boundary {\partial\Omega} consists of a discrete set of eigenvalues of the form: {\lambda_1<\lambda_2\leq \dots}

In this setting, the fundamental gap {\lambda_2-\lambda_1} of {-\Delta+V} is bounded from below by

\displaystyle \lambda_2 - \lambda_1 \geq 3\frac{\pi^2}{D^2}

Remark 1 This theorem is sharp: {\lambda_2-\lambda_1=3\frac{\pi^2}{D^2}} when {\Omega=(-D/2, D/2)\subset\mathbb{R}} and {V\equiv 0} (by Fourier analysis). In other terms, Andrews-Clutterbuck theorem is an optimal comparison 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 {S^3=\{(x_1,\dots, x_4\in\mathbb{R}^4: x_1^2+\dots+x_4^2=1\}} is isometric to Clifford torus {\mathbb{T}=\{(x_1,\dots,x_4)\in\mathbb{R}^4: x_1^2+x_2^2 = x_3^2+x_4^2 = 1/2\}}.

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 {\gamma=\lambda_2-\lambda_1} 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

\displaystyle \frac{\partial}{\partial t} u(t,x) = \Delta u(t,x) - V(x)u(t,x), \quad (t,x)\in [0,\infty)\times \Omega

with {u\equiv 0} on {\partial \Omega}. More precisely, one has that

\displaystyle u(t,x) = c \exp(\lambda_1t) f_1(x) (1+O(\exp(-\gamma t))


  • {c} is an adequate constant,
  • {f_1} is the ground state of {-\Delta+V}, i.e., {-\Delta f_1+V f_1=\lambda_1 f_1}, {f_1>0} on {\textrm{int}(\Omega)}, {f_1=0} on {\partial\Omega} and {f_1} is normalized so that {\int_{\Omega} f_1^2=1}, and
  • {O(\exp(-\gamma t))} denotes (as usual) a quantity bounded from above by {C\exp(-\gamma t)} for some constant {C>0} and all {t\geq 0}.

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 {\lambda_2-\lambda_1\geq 3\pi^2/D^2} for the free Laplacian ({V\equiv 0}) on several convex domains. After that, Singer-Wong-Yau-Yau proved that

\displaystyle \lambda_2-\lambda_1\geq \frac{1}{4}\left(\frac{\pi^2}{D^2}\right)

and Yu-Zhong improved this result by showing that

\displaystyle \lambda_2-\lambda_1\geq \frac{\pi^2}{D^2}

Furthermore, some particular cases of Andrews-Clutterbuck were previously known: for instance, Lavine proved the one-dimensional case {\Omega\subset\mathbb{R}}, 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 {\Sigma} without boundary can be minimally embedded in {S^3}.

Remark 2 The analog of Lawson’s theorem is completely false in {\mathbb{R}^3}: using the maximum principle, one can show that there are no immersed compact minimal surfaces in {\mathbb{R}^3}.

Moreover, Lawson (in the same paper loc. cit.) showed that, if the genus of {\Sigma} is not prime, then {\Sigma} admits two non-isometric minimal embeddings in {S^3}.

On the other hand, Lawson’s construction in the case of genus {1} produces only the Clifford torus (up to isometries). Nevertheless, Lawson proved (in this paper here) that if {\Sigma\subset S^3} is a minimal torus, then there exists a diffeomorphism {F:S^3\rightarrow S^3} taking {\Sigma} to the Clifford torus {\mathbb{T}}: in other terms, there is no knotted minimal torus in {S^3}!

In this context, Lawson was led to conjecture that this diffeomorphism {F:S^3\rightarrow S^3} could be taken to be an isometry, an assertion that was confirmed by Brendle.

Read More…

Simion Filip, Giovanni Forni and I have just upload to ArXiv our paper Quaternionic covers and monodromy of the Kontsevich-Zorich cocycle in orthogonal groups.

This article is motivated by Simion Filip’s recent work on the classification of possible monodromy groups for the Kontsevich-Zorich cocycle.

Very roughly speaking, the basic idea of this classification is the following. Consider the Kontsevich-Zorich cocycle on the Hodge bundle over the support of an ergodic {SL(2,\mathbb{R})}-invariant probability measure on (a connected component of) a stratum of the moduli spaces of translation surfaces. Recall that, in a certain sense, the Kontsevich-Zorich cocycle is a sort of “foliated monodromy representation” obtained by using the Gauss-Manin connection on the Hodge bundle while essentially moving only along {SL(2,\mathbb{R})}-orbits on moduli spaces of translation surfaces.

By extending a previous work of Martin Möller (for the Kontsevich-Zorich cocycle over Teichmüller curves), Simion Filip showed (in this paper here) that a version of the so-called Deligne’s semisimplicity theorem holds for the Kontsevich-Zorich cocycle: in plain terms, this means that the Kontsevich-Zorich cocycle can be completely decomposed into ({SL(2,\mathbb{R})}-)irreducible pieces, and, furthermore, each piece respects the Hodge structure coming from the Hodge bundle. In other terms, the Kontsevich-Zorich cocycle is always diagonalizable by blocks and its restriction to each block is related to a variation of Hodge structures of weight {1}.

The previous paragraph might seem abstract at first sight, but, as it turns out, it imposes geometrical constraints on the possible groups of matrices obtained by restriction of the Kontsevich-Zorich cocycle to an irreducible piece. More precisely, by exploiting the known tables (see § 3.2 of Filip’s paper) for monodromy representations coming from variations of Hodge structures of weight {1} over quasiprojective varieties, Simion Filip classified (up to compact and finite-index factors) the possible Zariski closures of the groups of matrices associated to restrictions of the Kontsevich-Zorich cocycle to an irreducible piece. In particular, there are at most five types of possible Zariski closures for blocks of the Kontsevich-Zorich cocycle (cf. Theorems 1.1 and 1.2 in Simion Filip’s paper):

Moreover, each of these items can be realized as an abstract variation of Hodge structures of weight {1} over abstract curves and/or Abelian varieties.

Here, it is worth to stress out that Filip’s classification of the possible blocks of the Kontsevich-Zorich cocycle comes from a general study of variations of Hodge structures of weight {1}. Thus, it is not clear whether all items above can actually be realized as a block of the Kontsevich-Zorich cocycle over the closure of some {SL(2,\mathbb{R})}-orbit in the moduli spaces of translations surfaces.

In fact, it was previously known in the literature that (all groups listed in) the items (i) and (ii) appear as blocks of the Kontsevich-Zorich cocycle (over closures of {S(2,\mathbb{R})}-orbits of translation surfaces given by certain cyclic cover constructions). On the other hand, it is not obvious that the other 3 items occur in the context of the Kontsevich-Zorich cocycle, and, indeed, this realizability question was explicitly posed by Simion Filip in Question 5.5 of his paper (see also § B.2 in Appendix B of this recent paper of Delecroix-Zorich).

In our paper, Filip, Forni and I give a partial answer to this question by showing that the case {SO^*(6)} of item (iv) is realizable as a block of the Kontsevich-Zorich cocycle.

Remark 1 Thanks to an exceptional isomorphism between the real Lie algebra {\mathfrak{so}^*(6)} in its standard representation and the second exterior power representation of the real Lie algebra {\mathfrak{su}(3,1)}, this also means that the case of {\wedge^2 SU(3,1)} of item (iii) is also realized.

Remark 2 We think that the examples constructed in this paper by Yoccoz, Zmiaikou and myself of regular origamis associated to the groups {SL(2,\mathbb{F}_p)} of Lie type might lead to the realizability of all groups {SO^*(2n)} in item (iv). In fact, what prevents Filip, Forni and I to show that this is the case is the absence of a systematic method to show that the natural candidates to blocks of the Kontsevich-Zorich cocycle over these examples are actually irreducible pieces.

In the remainder of this post, we will briefly explain our construction of an example of closed {SL(2,\mathbb{R})}-orbit such that the Kontsevich-Zorich cocycle over this orbit has a block where it acts through a Zariski dense subgroup of {SO^*(6)} (modulo compact and finite-index factors).

Read More…

Posted by: matheuscmss | February 22, 2015

First Bourbaki seminar of 2015 (I): Harari’s talk

About one month ago (on January 24, 2015), the first Bourbaki seminar of 2015 took place at Institut Henri Poincaré. As usual, this was an excellent opportunity to learn about recent advances in areas of Mathematics outside my field of expertise.

The first Bourbaki seminar of 2015 had the following four talks:

Today, I would like to discuss David Harari’s talk entitled “Zero cycles and rational points on fibrations in rationally connected varieties (after Harpaz and Wittenberg)”. Here, I will try to follow the first 38m50s of the video of Harari’s talk (in French) and sometimes his lecture notes (also in French). Of course, this goes without saying that any errors/mistakes are my full responsibility.

1. Introduction

One of the basic old problems in Number Theory is to determine whether a system of polynomial equations

\displaystyle P_i(x_1,\dots, x_n) = 0, \quad 1\leq i\leq r \ \ \ \ \ (1)

associated to homogeneous polynomials {P_i} with coefficients in a number field {k} has non-trivial solutions.

Equivalently, denoting by {X} the algebraic variety defined by the system (1), we want to know whether the set {X(k)} of points of {X} whose coordinates belong to {k} is not empty. In the literature, {X(k)} is called the set of {k}rational points of {X}.

It is not easy to answer this problem in general. Nevertheless, we have the following necessary condition: if {X(k)\neq\emptyset}, then {X(k_v)\neq\emptyset} for all completion {k_v} of {k} with respect to a place of {k} (i.e., {v} is an equivalence class of absolute values). In other words, we have that {X(k)=\emptyset} whenever there is a local obstruction in the sense that {X(k_v)=\emptyset} for some place {v} of {k}.

This necessary condition based on local obstructions is helpful because it is often easy to verify algorithmically that {X(k_v)\neq\emptyset}. For example, when {k=\mathbb{Q}}, its completions {k_v} are either {k_v=\mathbb{Q}_p} (for the place of {p}-adic absolute values, {p\in\mathbb{N}} prime) or {k_v=\mathbb{R}=\mathbb{Q}_{\infty}} (for the “place at infinity”), and, in this situation, we can check that {X(\mathbb{Q}_p)\neq\emptyset} with the help of Hensel’s lemma ({p}-adic analog of Newton’s method).

It is known that this necessary condition is sufficient in certain special cases. For instance, the classical Hasse-Minkowski theorem (from 1924) states that {X(k)\neq\emptyset} if and only if {X(k_v)\neq\emptyset} when {X} is a quadric, i.e., {X} is defined by just one polynomial equation of degree {2}.

Partly motivated by this, we introduce the following definition:

Definition 1 {X} satisfies Hasse’s principle (also called local-global principle) whenever {X(k)\neq\emptyset} if and only if {X(k_v)\neq\emptyset} for all places {v} of {k}.

As it turns out, Hasse’s principle is false in general: Swinnerton-Dyer constructed in 1962 some counterexamples among cubic surfaces, and Iskovskih constructed in 1970 a counterexample among the surfaces fibered in conics (given by intersections of two projective quadrics).

Of course, given that it is not hard to determine algorithmically when {X(k_v)\neq\emptyset} (with the help of Hensel lemma and/or Newton’s method), it is somewhat sad that Hasse’s principle fails in general.

In view of this state of affairs, we can try to generalize the problem of determining whether {X(k)\neq\emptyset} by replacing “rational points” by slightly more general objects (which then would be easier to find). In this direction, we have the following notion.

Definition 2 A zero-cycle {z} is a formal linear combination {z=\sum\limits_{x} n_x x} where:

  • {n_x\in\mathbb{Z}} vanishes for all but finitely many {x\in X}, and
  • if {n_x\neq 0}, then {x} is a closed point in the sense of Algebraic Geometry, i.e., {x} is a point defined over (its coordinates belong to) a finite extension {k(x)} of {k}.

The degree {deg(z)} of a zero-cycle {z} is {deg(z):=\sum\limits_{x} n_x [k(x):k]}.

Note that, by definition, a rational point {x\in X(k)} is a zero-cycle {z=x} of degree {deg(z) = 1}. Thus, we can ask the following more general question:

Does {X} possess a zero-cycle of degree {1} if {X} has such cycles over all {k_v}?

Remark 1 It follows from Bézout’s theorem that {X} has a zero-cycle of degree {1} if and only if {X} has points defined over finite extensions of {k} whose degrees are coprime.

Remark 2 A little curiosity about Bézout: as I discovered after moving from Paris to Avon, Bézout spent the last years of his life in Avon and the city gave his name to a street (not far from my appartment) in his honor.

Once more, the answer to this question is no: for example, it is known that there are counterexamples among surfaces fibered in conics.

Given this scenario, our goal is to explain how to refine the local-global principle with additional cohomological conditions (related to the so-called Brauer groups) introduced by Manin ensuring the existence of zero-cycles and/or rational points in certain situations.

Read More…

Posted by: matheuscmss | November 15, 2014

On the ergodicity of billiards in non-rational polygons

A couple of days ago (on November 12th, 2014 to be more precise), Giovanni Forni gave a talk at the “flat seminar / séminaire plat” on the ergodicity of billiards on non-rational polygons, and, by following the suggestion of two friends, I will transcript in this post my notes from Giovanni’s talk.

[Update (November 20, 2014): Some phrases near the statement of Theorem 3 below were edited to correct an inaccuracy pointed out to me by Giovanni.]

Let {\mathcal{P}\subset \mathbb{R}^2} be a polygon with {d+1} sides and denote by {\theta_1, \dots, \theta_{d+1}} its interior angles.

The billiard flow associated to {\mathcal{P}} is the following dynamical system. A point-particle in {\mathcal{P}} follows a linear trajectory with unit speed until it hits the boundary of {\mathcal{P}}. At such an instant, the point-particle is reflected by the boundary of {\mathcal{P}} (according to the usual laws of a specular reflection) and then it follows a new linear trajectory with unit speed. (Of course, this definition makes no sense at the corners of {\mathcal{P}}, and, for this reason, we leave the billiard flow undefined at any orbit going straight into a corner)

The phase space of the billiard flow is naturally identified with the three-dimensional manifold {\mathcal{P}\times S^1}: indeed, we need an element of {\mathcal{P}} to describe the position of the particle and an element of the unit circle {S^1\subset \mathbb{R}^2} to describe the velocity vector of the particle.

Alternatively, the billiard flow associated to {\mathcal{P}} can be interpreted as the geodesic flow on a sphere {S^2} with a flat metric and {(d+1)} conical singularities (whose cone angles are {2\theta_1, \dots, 2\theta_{d+1}}) with non-trivial holonomy (see Section 2 of Zorich’s survey): roughly speaking, one obtains this flat sphere with conical singularities by taking two copies of {\mathcal{P}} (one on the top of the other), gluing them along the boundaries, and by thinking of a billiard flow trajectory on {\mathcal{P}} as a straight line path going from one copy of {\mathcal{P}} to the other at each reflection.

This interpretation shows us that billiard flows on polygons are a particular case of geodesic flows {\{G_t\}} on the unit tangent bundle {S(M-\Sigma)} of compact flat surfaces {M} whose subsets {\Sigma} of conical singularities were removed.

Remark 1 In the case of a rational polygon {\mathcal{P}} (i.e., {\theta_1, \dots, \theta_{d+1}} are rational multiples of {\pi}), it is often a better idea (see this survey of Masur and Tabachnikov) to take several copies of {\mathcal{P}} obtained by applying the finite group generated by the reflections through the sides of {\mathcal{P}} and then glue by translation the pairs of parallel sides of the resulting figure. In this way, one obtains that the billiard flow associated to {\mathcal{P}} is equivalent to translation (straightline) flow on a translation surface (an object that has trivial holonomy and, hence, is more well-behaved that a flat metric on {S^2} with conical singularities) and this partly explains why the Ergodic Theory of billiards on rational polygons is well-developed. However, let us not insist on this point here because in what follows we will be mostly interested in billiard flows on irrational polygons.

A basic problem concerning the dynamics of billiards flows on polygons, or, more generally, geodesic flows on flat surfaces with conical singularities is to determine whether such a dynamical system is ergodic.

In view of Remark 1, we can safely skip the case of rational polygons: indeed, this setting one can use the relationship to translation surfaces to give a satisfactory answer to this problem (see the survey of Masur and Tabachnikov for more explanations). So, from now on, we will focus on billiard flows associated to non-rational polygons.

Kerckhoff, Masur and Smillie proved in 1986 that the billiard flow is ergodic for a {G_{\delta}}-dense subset of polygons. Their idea is to consider the {G_{\delta}}-dense subset of “Liouville polygons” admitting fast approximations by rational polygons (i.e., the subset of polygons whose interior angles admit fast approximations by rational multiples of {\pi}). Because the ergodicity of the billiard flow on rational polygons is well-understood, one can hope to “transfer” this information from rational polygons to any “Liouville polygon”.

Remark 2 The {G_{\delta}}-dense subset of polygons constructed by Kerckhoff, Masur and Smillie has zero measure: indeed, this happens because they require the angles {\theta_1,\dots, \theta_{d+1}} to be “Liouville” (i.e., admit fast approximations by rational multiples of {\pi}), and, as it is well-known, the subset of Liouville numbers has zero Lebesgue measure.

A curious feature of the argument of Kerckhoff, Masur and Smillie is that it is hard to extract any sort of quantitative criterion. More precisely, it is difficult to quantify how fast the quantities {\theta_1/\pi, \dots, \theta_{d+1}/\pi} must be approximated by rationals in order to ensure that the ergodicity of the billiard flow on the corresponding polygon. This happens because the genera of translation surfaces associated to the rational polygons approximating {\theta_1,\dots,\theta_{d+1}} usually tend to infinity and it is a non-trivial problem to control the ergodic properties of translation flows on families of translation surfaces whose genera tend to infinity.

Nevertheless, Vorobets obtained in 1997 (by other methods) a quantitative version of Kerckhoff, Masur and Smillie by showing the ergodicity of the billiard flow on a polygon {\mathcal{P}} whose interior angles {\theta_1,\dots,\theta_{d+1}} verify the following fast approximation property: there exist arbitrarily large natural numbers {N\in \mathbb{N}} such that

\displaystyle \left|\theta_{1}-\pi \frac{p_1}{q_1}\right|<\left(2^{2^{2^{2^N}}}\right)^{-1}, \dots, \left|\theta_{d+1}-\pi \frac{p_{d+1}}{q_{d+1}}\right|<\left(2^{2^{2^{2^N}}}\right)^{-1}

for some rational numbers {p_i/q_i\in\mathbb{Q}}, {i=1,\dots, d+1}, with denominators {q_i\leq N}, {i=1,\dots, N}.

In summary, the works of Kerckhoff-Masur-Smillie and Vorobets allows to solve the problem of ergodicity of the billiard flow on Liouville polygons.

Of course, this scenario motivates the question of ergodicity of billiard flows on Diophantine polygons (i.e., the “complement” of Liouville polygons consisting of those {\mathcal{P}} which are badly approximated by rational polygons).

In his talk, Giovanni announced a new criterion for the ergodicity of the billiard flow on polygons (and, more generally, the geodesic flow on a flat surface with conical singularities) with potential applications to a whole class (of full measure) of Diophantine polygons.

Read More…

« Newer Posts - Older Posts »



Get every new post delivered to your Inbox.

Join 150 other followers