Sometime ago, Alex Eskin and Alex Wright told me about the following question posed to them by Peter Sarnak:

How frequent are thin groups among Kontsevich-Zorich monodromies?

Instead of explaning the meaning of Sarnak’s question in general, we shall restrict ourselves to the case of Kontsevich-Zorich (KZ) monodromies associated to square-tiled surfaces.

More concretely, let {X=(M,\omega)} be a square-tiled surface (also called origamis) of genus {g\geq 1}, i.e., {p:M\rightarrow\mathbb{R}^2/\mathbb{Z}^2} is a finite branched covering which is unramified off {0\in\mathbb{R}^2/\mathbb{Z}^2} and {\omega = p^*(dz)} is the pullback of {dz=dx+i dy} on {\mathbb{R}^2/\mathbb{Z}^2 \simeq \mathbb{C}/\mathbb{Z}\oplus i\mathbb{Z}}. We have a natural representation

\displaystyle \rho_X:\textrm{Aff}(X)\rightarrow Sp(H_1^{(0)}(X,\mathbb{Z}))\simeq Sp(2g-2,\mathbb{Z})

from the group {\textrm{Aff}(X)} of affine homeomorphisms of {X} to the group {Sp(H_1^{(0)}(X,\mathbb{Z}))} of symplectic matrices of the subspace {H_1^{(0)}(X,\mathbb{Z})} of integral homology classes of {X} projecting to zero under {p}. In this setting, the Kontsevich-Zorich monodromy {\Gamma_X} (associated to the {SL(2,\mathbb{R})}-orbit of {X} in the moduli space of translation surfaces) is the image of {\rho_X}, i.e.,

\displaystyle \Gamma_X := \rho_X(\textrm{Aff}(X))

(See e.g. these posts here for more background material on square-tiled surfaces.)

By following Sarnak’s terminology, we will say that {\Gamma_X} is a thin group if {\Gamma_X} is an infinite index subgroup of {Sp(2g-2,\mathbb{Z})} whose Zariski closure is

\displaystyle \overline{\Gamma_X}^{\textrm{Zariski}} = Sp(2g-2,\mathbb{R})

In the particular case of square-tiled surfaces, Sarnak’s question above is related to the following two problems:

  • (a) find examples of square-tiled surfaces {X} whose KZ monodromies {\Gamma_X} are thin;
  • (b) decide whether the “majority” of square-tiled surfaces in a given connected component {\mathcal{C}} of a stratum of the moduli spaces of unit area translation surfaces has thin KZ monodromy (here, “majority” could mean “all but finitely many” or “almost full probability as the number of squares/tiles grows”.)

The goal of this post is to record (below the fold) some discussions with Vincent Delecroix and certain participants of MathOverFlow around item (a).

Remark 1 While we will not give answers to items (a) and/or (b) in this post, we decided to write it down anyway with the hope that it might be of interest to some readers of this blog: in fact, by the end of this post, we will show the following conditional statement: if the group generated by the matrices

\displaystyle A=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right), \quad B=\left(\begin{array}{cccc} -1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)

has infinite-index in {Sp(4,\mathbb{Z})}, then a certain square-tiled surface of genus {3} answers item (a) affirmatively.

Remark 2 Some “evidence” supporting a positive answer to item (b) is provided by this recent paper of Fuchs-Rivin where it is shown that two “randomly chosen” elements (in {SL(n,\mathbb{Z})}) “tend” to generate thin groups.

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Posted by: matheuscmss | November 18, 2015

Harmonic quasi-isometries (after Benoist and Hulin)

Last September 28, Yves Benoist gave a beautiful talk on the occasion of the workshop Geometry and Dynamics on Moduli Spaces (that is, one of the four 2015 Clay Research Workshops) about his joint work with Dominique Hulin on a generalized version of the so-called Schoen-Li-Wang conjecture (on harmonic maps within bounded distance to given quasi-isometries of symmetric spaces of rank one).

Remark 1 Still concerning beautiful talks delivered in this conference, I strongly recommend taking a look at this video of Peter Scholze’s talk on cohomology of algebraic varieties: indeed, I think that he was extremely sucessful in communicating his results to a broad audience of non-experts in Algebraic Geometry (such as myself).

This post is a transcription of my notes for Yves’ lecture and, as usual, all errors/mistakes below are my entire responsibility.

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Last August 25th, Harald Helfgott gave the talk “Soficity, short cycles and the Higman group” (based on joint work with Kate Juschenko) during the Second Workshop on Combinatorics, Number Theory and Dynamical Systems coorganized by Christian Mauduit, Carlos Gustavo Moreira, Yuri Lima and myself at IMPA (Brazil).

This post is a transcription of my notes for Harald’s talk, and, evidently, all mistakes/errors are my responsibility.

1. Some notations

We denote by {Sym(n)} the group of all permutations of {\{1,\dots, n\}}.

For {g_1, g_2\in Sym(n)}, we define their distance {d_n(g_1, g_2)} by

\displaystyle d_n(g_1, g_2):=\frac{1}{n}\#\{1\leq i\leq n: g_1(i)\neq g_2(i)\}

Definition 1 (Gromov; Weiss) Let {G} be a group. Given {\delta>0} and {S\subset G} finite, we say that {\phi:S\rightarrow Sym(n)} is a {(S,n,\delta)}-sofic representation whenever

  • (a) {\phi} is an “approximate homomorphism”: {d_n(\phi(g)\phi(h),\phi(gh))<\delta} for all {g,h\in S} with {gh\in S};
  • (b) {\phi(g)\in Sym(n)} has “few” fixed points for {g\neq e}: {d_n(\phi(g),id)>1-\delta} for all {g\in S-\{e\}}.

We say that a group {G} is sofic if it has {(S,n,\delta)}-sofic representation for all {S\subset G} finite, all {\delta>0} (and some {n\in\mathbb{N}}).

Basic examples of sofic groups are: finite groups, amenable groups, etc. In general, it is known that several families of groups are sofic, but it is an important open problem to construct (or show the existence of) non-sofic groups.

The goal of this post is to discuss a candidate for non-sofic group and its connections to Number Theory.

1.1. Higman groups

For {m\geq 2}, let

\displaystyle H_m := \langle a_1,\dots, a_m : a_i^{-1} a_{i+1} a_i = a_{i+1}^2 \textrm{ for all } i\in\mathbb{Z}/m\mathbb{Z} \rangle

The groups {H_2} and {H_3} are trivial, and the group {H_4} is the so-called Higman group.

Remark 1 Several statements in this post can be generalized for {H_m} for all {m\geq 4}, but for the sake of exposition we will stick to {H_4}.

Theorem 2 (Helfgott-Juschenko) Assume that {H_4} is sofic.Then, for every {\varepsilon>0}, there exists {n\in\mathbb{N}} and a bijection {f:\mathbb{Z}/n\mathbb{Z}\rightarrow\mathbb{Z}/n\mathbb{Z}} such that

  • (a) {f} is an “almost exponential function”: {f(x+1)=2f(x)} for all {x\in S} where {S\subset\mathbb{Z}/n\mathbb{Z}} is a subset of cardinality {|S|\geq (1-\varepsilon)n}.
  • (b) {f^4(x):=f\circ f\circ f\circ f(x)=x} for all {x\in\mathbb{Z}/n\mathbb{Z}}.

Remark 2 The existence of functions {f} as above is “unlikely” when {\varepsilon>0} is small. More precisely, it is possible to show that there are no bijections {f:\mathbb{Z}/n\mathbb{Z}\rightarrow\mathbb{Z}/n\mathbb{Z}} satisfying item (a) with {\varepsilon=0} and item (b) when {n=p^5} is the fifth power of a prime {p} (cf. Remark 4 below for a more precise statement).

In other words, if we could take {\varepsilon=0} and {n=p^5} in the statement of Theorem 2, the non-soficity of the Higman group {H_4} would follow.

Unfortunately, the techniques of Helfgott and Juschenko do not allow us to take {\varepsilon=0} in Theorem 2, but they permit to control the integer {n}. More concretely, as we are going to see in Theorem 4 below, the integer {n} can be chosen from any fixed sequence {0<n_1<n_2<\dots} which is thick in the following sense:

Definition 3 A sequence {0<n_1<n_2<\dots} of positive integers is thick if for every {\varepsilon>0} there exists {K>0} such that

\displaystyle \frac{n_{k+1}}{n_k}\leq 1+\varepsilon

for all {k>K}.

Remark 3 It does not take much to be a thick sequence: for example, the sequences {\{n^3\}_{n\in\mathbb{N}}} and {\{p^5\}_{p \textrm{ prime}}} are thick.

As we already announced, the main result of Helfgott-Juschenko is the following improvement of Theorem 2:

Theorem 4 (Helfgott-Juschenko) Assume that {H_4} is sofic and let {0<n_1<n_2<\dots} be a thick sequence.Then, for every {\varepsilon>0}, there exists {k} and a bijection {f:\mathbb{Z}/n_k\mathbb{Z}\rightarrow\mathbb{Z}/n_k\mathbb{Z}} such that

  • (a) {f(x+1)=2f(x)} for all {x\in S} where {S\subset\mathbb{Z}/n_k\mathbb{Z}} is a subset of cardinality {|S|\geq (1-\varepsilon)n_k}.
  • (b) {f^4(x):=f\circ f\circ f\circ f(x)=x} for all {x\in\mathbb{Z}/n_k\mathbb{Z}}.

Remark 4 The non-soficity of Higman group {H_4} would follow from this theorem if the bijection {f:\mathbb{Z}/n_k\mathbb{Z}\rightarrow \mathbb{Z}/n_k\mathbb{Z}}, {n_k\in\{p^5\}_{p\textrm{ prime}}}, provided by this statement could be taken so that

  • (a*) {f(x+1)=2f(x)} for all {x\in S} where {S\subset\mathbb{Z}/n_k\mathbb{Z}} is a subset of cardinality {|S|\geq n_k-n_k^{1/6}}.
  • (b) {f^4(x):=f\circ f\circ f\circ f(x)=x} for all {x\in\mathbb{Z}/n_k\mathbb{Z}}.

Indeed, this is so because Glebsky and Holden-Robinson proved that there is no {f} verifying (a*) and (b).

Remark 5 A natural question related to Theorem 4 is: what happens with fewer iterations in item (b)? In this situation, it is possible to use the fact that the group {H_3} is trivial to show that, for each {\varepsilon>0}, there exists {n_{\varepsilon}\in\mathbb{N}} such that for any {n\geq n_{\varepsilon}} there is no bijection {f:\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}} such that

  • (a) {f(x+1)=2f(x)} for all {x\in S} for {|S|\geq (1-\varepsilon)n}.
  • (b) {f^3(x):=f\circ f\circ f(x)=x} for all {x\in S} for {|S|\geq (1-\varepsilon)n}.

This last remark can be generalized as follows.

Theorem 5 (Helfgott-Juschenko) Let {n, g\geq 2} be coprime. Consider the function {f:\{0,1,\dots, n-1\}\rightarrow\{0,1,\dots, n-1\}} given by

\displaystyle f(x)=g^x (\textrm{mod } n)

Then, the equation

\displaystyle f^3(x):=f\circ f\circ f(x)=x

has at most {o(n)} solutions {x\in\{0,1,\dots, n-1\}}.

Remark 6 This theorem improves on a result of Glebsky-Shparlinski.

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Posted by: matheuscmss | September 27, 2015

Third Bourbaki seminar of 2015: Sophie Morel

Three months ago, Sophie Morel gave her Bourbaki seminar talk “Construction de représentations galoisiennes [d’après Scholze]”.

As it turns out, Sophie Morel was kind towards the non-experts in this subject (like myself): indeed, a large part of the talk (see the video here) was introductory, while the more advanced material was delegated to the lecture notes (available here).

In the remainder of this post, I’ll try to summarize some of the topics discussed by Sophie Morel’s talk (using the video in the link above as the main source).

Disclaimer: Since I’m not an expert on this subject, all mistakes in this post are my responsibility.

Remark 1 If you are in the Oxford area this week, then you will have the opportunity to learn about the main results in Morel’s talk directly from the authors: indeed, Boxer and Scholze (resp.) will give talks tomorrow, resp. Wednesday on their works (see this schedule here of the corresponding 2015 Clay research conference.)

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Consider a random product of two symplectic matrices {A_0, A_1 \in Sp(V)} on a real symplectic vector space {V} of dimension {\textrm{dim}(V)=2d}, that is, the (symplectic) linear cocycle

\displaystyle F: \{0,1\}^{\mathbb{Z}}\times V\rightarrow \{0,1\}^{\mathbb{Z}}\times V

given by

\displaystyle F(x,v) = (\sigma(x), A_{x_0}v)

where {x=(x_n)_{n\in\mathbb{Z}}} and {\sigma(x):=(x_{n+1})_{n\in\mathbb{Z}}} is the shift map equipped with the Bernoulli measure {\mathbb{P}=(\frac{1}{2}\delta_0 + \frac{1}{2}\delta_1)^{\mathbb{Z}}}.

By Oseledets multiplicative ergodic theorem, the Lyapunov exponents of the random product of {A_0} and {A_1} (i.e., the linear cocycle {F}) are well-defined quantities {\lambda_1\geq\dots\geq\lambda_{2d}} (depending only on {A_0} and {A_1}) describing the exponential growth of the singular values of the random products

\displaystyle A_{x_{m}}\dots A_{x_0}, \quad m\in\mathbb{N}

for any {\mathbb{P}}-typical choice of {x=(x_n)_{n\in\mathbb{Z}}}.

Moreover, the fact that {A_0} and {A_1} are symplectic matrices implies that the Lyapunov exponents are symmetric with respect to the origin, i.e., {\lambda_{2d-k-1} = -\lambda_k} for each {k=1,\dots,d}. In other words, the Lyapunov exponents of the symplectic linear cocycle {F} have the form:

\displaystyle \lambda_1\geq\dots\geq\lambda_d\geq-\lambda_d\geq\dots\geq-\lambda_1

In fact, this structure of the Lyapunov exponents of a symplectic linear cocycle reflects the fact that if {\theta} is an eigenvalue of a symplectic matrix {B}, then {\theta^{-1}} is also an eigenvalue of {B}.

A natural qualitative question about Lyapunov exponents concerns their simplicity in the sense that there are no repeated numbers in the list above (i.e., {\lambda_j > \lambda_{j+1}} for all {k=1,\dots,d}).

The simplicity property for Lyapunov exponents is the subject of several papers in the literature: see, e.g., the works of Furstenberg, Goldsheid-Margulis, Guivarch-Raugi, and Avila-Viana (among many others).

Very roughly speaking, the basic philosophy behind these papers is that the simplicity property holds whenever the monoid {\mathcal{M}} generated by {A_0} and {A_1} is rich. Of course, there are several ways to formalize the meaning of the word “rich”, for example:

  • Goldsheid-Margulis and Guivarch-Raugi asked {\mathcal{M}} to be Zariski-dense in {Sp(V)};
  • Avila-Viana required {\mathcal{M}} to be
    • pinching: there exists {C\in\mathcal{M}} whose eigenvalues are all real with distinct moduli; such a {C} is called a pinching matrix;
    • twisting: there exists a pinching matrix {C\in\mathcal{M}} and a twisting matrix {D\in\mathcal{M}} with respect to {C} in the sense that {D(F)\cap F'=\{0\}} for all isotropic {C}-invariant subspaces {F} and all coisotropic {C}-invariant subspaces {F'} with {\textrm{dim}(F) + \textrm{dim}(F')=2d}.

Of course, these notions of “richness” of a monoid {\mathcal{M}} are “close” to each other, but they differ in a subtle detail: while the Zariski-density condition on {\mathcal{M}} is an algebraic requirement, the pinching and twisting condition on {\mathcal{M}} makes no reference to the algebraic structure of the linear group {Sp(V)}.

In particular, this leads us to the main point of this post:

How the Zariski-density and pinching and twisting conditions relate to each other?

The first half of this question has a positive answer: a Zariski-dense monoid {\mathcal{M}} is also pinching and twisting. Indeed:

  • (a) a modification of the arguments in this blog post here (in Spanish) permits to prove that any Zariski-dense monoid {\mathcal{M}} contains a pinching matrix {C}, and
  • (b) the twisting condition on a matrix {D} with respect to a pinching matrix {C} can be phrased in terms of the non-vanishing of certain (isotropic) minors of the matrix of {D} written in a basis of eigenvectors of {C}; thus, a Zariski-dense monoid {\mathcal{M}} contains a twisting matrix with respect to any given pinching matrix.

On the other hand, the second half of this question has a negative answer: we exhibit below a pinching and twisting monoid {\mathcal{M}} which is not Zariski dense.

Remark 1 The existence of such examples of monoids is certainly known among experts. Nevertheless, I’m recording it here because it partly “justifies” a forthcoming article joint with Artur Avila and Jean-Christophe Yoccoz in the following sense.

The celebrated paper of Avila-Viana quoted above (on Kontsevich-Zorich conjecture) shows that the so-called “Rauzy monoids” are pinching and twisting (and this is sufficient for their purposes of proving simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle for Masur-Veech measures).

On the other hand, since a pinching and twisting monoid is not necessarily Zariski dense (as we are going to see below), the results of Avila-Viana (per se) can not answer a question of Zorich (see also Remark 6.12 in Avila-Viana paper) about the Zariski density of Rauzy monoids.

In this direction, Artur, Jean-Christophe and I solve (in an article still in preparation) Zorich’s question about Zariski density of Rauzy monoids in the special case of hyperelliptic Rauzy diagrams, and the main example of this post (which will be included in our forthcoming article with Artur and Jean-Christophe) serves to indicate that the results obtained by Artur, Jean-Christophe and myself can not be deduced as “abstract consequences” of the arguments in Avila-Viana paper.

Remark 2 The main example of this post also shows that (a version of) Prasad-Rapinchuk’s criterion for Zariski density (cf. Theorem 9.10 of Prasad-Rapinchuk paper or Theorem 1.5 in Rivin’s paper) based on Galois-pinching (in the sense of this paper here) and twisting properties is “sharp”: indeed, an important feature of the main example of this post is the failure of the Galois-pinching property (cf. Remark 4 below for more comments).

1. A monoid of 4×4 symplectic matrices

Let {\rho} be the third symmetric power of the standard representation of {SL(2,\mathbb{R})}. In concrete terms, {\rho} is constructed as follows. Consider the basis {\mathcal{B} = \{X^3, X^2Y, XY^2, Y^3\}} of the space {V} of homogenous polynomials of degree {3} on two variables {X} and {Y}. By letting {g=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\in SL(2,\mathbb{R})} act on {X} and {Y} as {g(X)=aX+cY} and {g(Y)=bX+dY}, we get a linear map {\rho(g)} on {V} whose matrix in the basis {\mathcal{B}} is

\displaystyle \rho\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) = \left(\begin{array}{cccc} a^3 & a^2 b & a b^2 & b^3 \\ 3 a^2 c & a^2 d + 2 a b c & b^2 c + 2 a b d & 3 b^2 d \\ 3 a c^2 & b c^2 + 2 a c d & a d^2 + 2 b c d & 3 b d^2 \\ c^3 & c^2 d & c d^2 & d^3 \end{array}\right)

Remark 3 The faithful representation {\rho} is the unique irreducible four-dimensional representation of {SL(2,\mathbb{R})}.

The matrices {\rho(g)} preserve the symplectic structure on {V} associated to the matrix

\displaystyle J = \left(\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & 1/3 & 0 \\ 0 & -1/3 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)

Indeed, a direct calculation shows that if {g=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)}, then

\displaystyle (\rho(g))^T\cdot J\cdot \rho(g) = \left(\begin{array}{cccc} 0 & 0 & 0 & -(a d - b c)^3 \\ 0 & 0 & \frac{(a d - b c)^3}{3} & 0 \\ 0 & - \frac{(a d - b c)^3}{3} & 0 & 0 \\ (a d - b c)^3 & 0 & 0 & 0 \end{array}\right)

where {(\rho(g))^T} stands for the transpose of {\rho(g)}.

Therefore, the image {H=\rho(SL(2,\mathbb{R}))} is a linear algebraic subgroup of the symplectic group {Sp(V)}, and the Zariski closure of the monoid {\mathcal{M}} generated by the matrices

\displaystyle A = \rho\left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right) = \left(\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 1\end{array}\right)


\displaystyle B = \rho\left(\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}\right) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ 3 & 2 & 1 & 0 \\ 1 & 1 & 1 & 1\end{array}\right)

is precisely {\overline{\mathcal{M}}^{Zariski}=H}.

Remark 4 Coming back to Remark 2, observe that {H} does not contain Galois-pinching elements of {Sp(V)} in the sense of this paper here (i.e., pinching elements of {Sp(V)} with integral entries whose characteristic polynomial has the largest possible Galois group for a reciprocal polynomial [namely, the hyperoctahedral group]) because its rank is {1}. Alternatively, a straightforward computation reveals that the characteristic polynomial of {\rho(g)} is

\displaystyle (x^2-\textrm{tr}(g)\det(g)x+\det(g)^3)\cdot (x^2 - \textrm{tr}(g)(\textrm{tr}(g)^2 - 3\det(g))x + \det(g)^3)

and, consequently, the eigenvalues of {\rho(g)} are

\displaystyle \frac{1}{2}\det(g)\left(\textrm{tr}(g)\pm\sqrt{\textrm{tr}(g)^2 - 4 \det(g)}\right),


\displaystyle \frac{1}{2}\left(\textrm{tr}(g)(\textrm{tr}(g)^2 - 3\det(g)) \pm (\textrm{tr}(g)^2 - \det(g)) \sqrt{\textrm{tr}(g)^2 - 4 \det(g)}\right)

In particular, since the characteristic polynomial of {\rho(g)} always splits, it is never the case that {\rho(g)} is Galois-pinching.

On the other hand, the element {A.B\in\mathcal{M}} is pinching because its eigenvalues are

\displaystyle 9+4\sqrt{5} > \frac{3+\sqrt{5}}{2} > \frac{3-\sqrt{5}}{2} > \frac{1}{9+4\sqrt{5}}

Also, the matrix {A\in\mathcal{M}} is twisting with respect to {A.B}. Indeed, the columns of the matrix

\displaystyle M = \left(\begin{array}{cccc} -\frac{1}{4} + \frac{(9 + 4 \sqrt{5})}{4} & 1 - \frac{(3 + \sqrt{5})}{2} & 1 - \frac{(3 - \sqrt{5})}{2} & -\frac{1}{4} + \frac{(9 - 4 \sqrt{5})}{4} \\ \frac{9}{8} + \frac{3(9 + 4 \sqrt{5})}{8} & -2 + \frac{(3 + \sqrt{5})}{2} & -2 + \frac{(3 - \sqrt{5})}{2} & \frac{9}{8} + \frac{3(9 - 4 \sqrt{5})}{8} \\ -\frac{15}{8} + \frac{3(9 + 4 \sqrt{5})}{8} & \frac{(3 + \sqrt{5})}{2} & \frac{(3 - \sqrt{5})}{2} & -\frac{15}{8} + \frac{3(9 - 4 \sqrt{5})}{8} \\ 1 & 1 & 1 & 1 \end{array}\right)

consist of eigenvectors of {A.B}. Thus, {T=M^{-1}\cdot A\cdot M} is the matrix of {A} in the corresponding basis of eigenvectors of {A.B}. Moreover, {A} is twisting with respect to {A.B} if and only if all entries of {T} and all of its {2\times 2} minors associated to isotropic planes are non-zero (cf. Lemma 4.8 in this paper here). Finally, this last fact is a consequence of the following exact calculation (see also the numerical approximations) for {T} and its matrix of {2\times 2} minors:

\displaystyle \begin{array}{rcl} T&=& \left(\begin{array}{cccc} \frac{8(5 + 2 \sqrt{5})}{25} & \frac{2(5 + 3 \sqrt{5})}{25} & \frac{(5 + \sqrt{5})}{25} & \frac{1}{( 5 \sqrt{5})} \\ -\frac{6(5 + 3 \sqrt{5})}{25} & \frac{2(5 + \sqrt{5})}{25} & \frac{7}{5 \sqrt{5}} & -\frac{3(-5 + \sqrt{5})}{25} \\ \frac{3(5 + \sqrt{5})}{25} & -\frac{7}{5 \sqrt{5}} & -\frac{2(-5 + \sqrt{5})}{25} & \frac{6(-5 + 3 \sqrt{5})}{25} \\ -\frac{1}{5 \sqrt{5}} & \frac{(5 - \sqrt{5})}{25} & \frac{2}{5} - \frac{6}{5 \sqrt{5}} & -\frac{(8(-5 + 2 \sqrt{5})}{25} \end{array}\right) \\ &=&\left(\begin{array}{cccc} 3.03108 & 0.936656 & 0.289443 & 0.0894427 \\ -2.80997 & 0.578885 & 0.626099 & 0.331672 \\ 0.868328 & -0.626099 & 0.221115 & 0.409969 \\ -0.0894427 & 0.110557 & -0.136656 & 0.168916 \end{array}\right) \end{array}


\displaystyle \begin{array}{rcl} & & 2\times 2 \textrm{ minors of } T = \\ & & \left(\begin{array}{cccccc} \frac{56}{25} + \frac{24}{5 \sqrt{5}} & \frac{32}{25} + \frac{16}{5 \sqrt{5}} & \frac{18}{25} + \frac{6}{5 \sqrt{5}} & \frac{6}{25} + \frac{2}{5 \sqrt{5}} & \frac{2}{25} + \frac{2}{5 \sqrt{5}} & \frac{1}{25} \\  -\frac{32}{25} - \frac{16}{5 \sqrt{5}} & \frac{6}{25} + \frac{2}{5 \sqrt{5}} & \frac{9}{25} + \frac{9}{5 \sqrt{5}} & \frac{3}{25} + \frac{3}{5 \sqrt{5}} & \frac{11}{25} & -\frac{2}{25} + \frac{2}{5 \sqrt{5}} \\  \frac{6}{25} + \frac{2}{5 \sqrt{5}} & -\frac{3}{25} - \frac{3}{5 \sqrt{5}} & \frac{13}{25} & -\frac{4}{25} & -\frac{3}{25} + \frac{3}{5 \sqrt{5}} & \frac{6}{25} - \frac{2}{5 \sqrt{5}} \\ \frac{18}{25} + \frac{6}{5 \sqrt{5}} & -\frac{9}{25} - \frac{9}{5 \sqrt{5}} & -\frac{36}{25} & \frac{13}{25} & -\frac{9}{25} + \frac{9}{5 \sqrt{5}} & \frac{18}{25} - \frac{6}{5 \sqrt{5}} \\  -\frac{2}{25} - \frac{2}{5 \sqrt{5}} & \frac{11}{25} & \frac{9}{25} - \frac{9}{5 \sqrt{5}} & \frac{3}{25} - \frac{3}{5 \sqrt{5}} & \frac{6}{25} - \frac{2}{5 \sqrt{5}} & -\frac{32}{25} + \frac{16}{5 \sqrt{5}} \\   \frac{1}{25} & \frac{2}{25} - \frac{2}{5 \sqrt{5}} & \frac{18}{25} - \frac{6}{5 \sqrt{5}} & \frac{6}{25} - \frac{2}{5 \sqrt{5}} & \frac{32}{25} - \frac{16}{5 \sqrt{5}} & \frac{56}{25} - \frac{24}{5 \sqrt{5}} \end{array}\right) = \\ & & \left(\begin{array}{cccccc} 4.38663 & 2.71108 & 1.25666 & 0.418885 & 0.258885 & 0.04 \\ -2.71108 & 0.418885 & 1.16498 & 0.388328 & 0.44 & 0.0988854 \\ 0.418885 & -0.388328 & 0.52 & -0.16 & 0.148328 & 0.0611146 \\ 1.25666 & -1.16498 & -1.44 & 0.52 & 0.444984 & 0.183344 \\ -0.258885 & 0.44 & -0.444984 & -0.148328 & 0.0611146 & 0.151084 \\ 0.04 & -0.0988854 & 0.183344 & 0.0611146 & -0.151084 & 0.0933747 \end{array}\right) \end{array}

In summary, the monoid {\mathcal{M}} is pinching and twisting, but not Zariski dense in {Sp(V)}.

About two weeks ago, Weixiao Shen gave the talk “Hausdorff dimension of the graphs of the classical Weierstrass functions” during the Third Palis-Balzan International Symposium on Dynamical Systems.

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

1. Introduction

In Real Analysis, the classical Weierstrass function is

\displaystyle W_{\lambda,b}(x) = \sum\limits_{n=0}^{\infty} \lambda^n \cos(2\pi b^n x)

with {1/b < \lambda < 1}.

Note that the Weierstrass functions have the form

\displaystyle f^{\phi}_{\lambda,b}(x) = \sum\limits_{n=0}^{\infty} \lambda^n \phi(b^n x)

where {\phi} is a {\mathbb{Z}}-periodic {C^2}-function.

Weierstrass (1872) and Hardy (1916) were interested in {W_{\lambda,b}} because they are concrete examples of continuous but nowhere differentiable functions.

Remark 1 The graph of {f^{\phi}_{\lambda,b}} tends to be a “fractal object” because {f^{\phi}_{\lambda,b}} is self-similar in the sense that

\displaystyle f^{\phi}_{\lambda, b}(x) = \phi(x) + \lambda f^{\phi}_{\lambda,b}(bx)

We will come back to this point later.

Remark 2 {f^{\phi}_{\lambda,b}} is a {C^{\alpha}}-function for all {0\leq \alpha < \frac{-\log\lambda}{\log b}}. In fact, for all {x,y\in[0,1]}, we have

\displaystyle \frac{f^{\phi}_{\lambda, b}(x) - f^{\phi}_{\lambda,b}(y)}{|x-y|^{\alpha}} = \sum\limits_{n=0}^{\infty} \lambda^n b^{n\alpha} \left(\frac{\phi(b^n x) - \phi(b^n y)}{|b^n x - b^n y|^{\alpha}}\right),

so that

\displaystyle \frac{f^{\phi}_{\lambda, b}(x) - f^{\phi}_{\lambda,b}(y)}{|x-y|^{\alpha}} \leq \|\phi\|_{C^{\alpha}} \sum\limits_{n=0}^{\infty}(\lambda b^{\alpha})^n:=C(\phi,\alpha,\lambda,b) < \infty

whenever {\lambda b^{\alpha} < 1}, i.e., {\alpha < -\log\lambda/\log b}.

The study of the graphs of {W_{\lambda,b}} as fractal sets started with the work of Besicovitch-Ursell in 1937.

Remark 3 The Hausdorff dimension of the graph of a {C^{\alpha}}-function {f:[0,1]\rightarrow\mathbb{R}} is

\displaystyle \textrm{dim}(\textrm{graph}(f))\leq 2 - \alpha

Indeed, for each {n\in\mathbb{N}}, the Hölder continuity condition

\displaystyle |f(x)-f(y)|\leq C|x-y|^{\alpha}

leads us to the “natural cover” of {G=\textrm{graph}(f)} by the family {(R_{j,n})_{j=1}^n} of rectangles given by

\displaystyle R_{j,n}:=\left[\frac{j-1}{n}, \frac{j}{n}\right] \times \left[f(j/n)-\frac{C}{n^{\alpha}}, f(j/n)+\frac{C}{n^{\alpha}}\right]

Nevertheless, a direct calculation with the family {(R_{j,n})_{j=1}^n} does not give us an appropriate bound on {\textrm{dim}(G)}. In fact, since {\textrm{diam}(R_{j,n})\leq 4C/n^{\alpha}} for each {j=1,\dots, n}, we have

\displaystyle \sum\limits_{j=1}^n\textrm{diam}(R_{j,n})^d\leq n\left(\frac{4C}{n^{\alpha}}\right)^d = (4C)^{1/\alpha} < \infty

for {d=1/\alpha}. Because {n\in\mathbb{N}} is arbitrary, we deduce that {\textrm{dim}(G)\leq 1/\alpha}. Of course, this bound is certainly suboptimal for {\alpha<1/2} (because we know that {\textrm{dim}(G)\leq 2 < 1/\alpha} anyway).Fortunately, we can refine the covering {(R_{j,n})} by taking into account that each rectangle {R_{j,n}} tends to be more vertical than horizontal (i.e., its height {2C/n^{\alpha}} is usually larger than its width {1/n}). More precisely, we can divide each rectangle {R_{j,n}} into {\lfloor n^{1-\alpha}\rfloor} squares, say

\displaystyle R_{j,n} = \bigcup\limits_{k=1}^{\lfloor n^{1-\alpha}\rfloor}Q_{j,n,k},

such that every square {Q_{j,n,k}} has diameter {\leq 2C/n}. In this way, we obtain a covering {(Q_{j,n,k})} of {G} such that

\displaystyle \sum\limits_{j=1}^n\sum\limits_{k=1}^{\lfloor n^{1-\alpha}\rfloor} \textrm{diam}(Q_{j,n,k})^d \leq n\cdot n^{1-\alpha}\cdot\left(\frac{2}{n}\right)^d\leq (2C)^{2-\alpha}<\infty

for {d=2-\alpha}. Since {n\in\mathbb{N}} is arbitrary, we conclude the desired bound

\displaystyle \textrm{dim}(G)\leq 2-\alpha

A long-standing conjecture about the fractal geometry of {W_{\lambda,b}} is:

Conjecture (Mandelbrot 1977): The Hausdorff dimension of the graph of {W_{\lambda,b}} is

\displaystyle 1<\textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b} < 2

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

\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) \geq 2 + \frac{\log\lambda}{\log b}

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

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

Theorem 1 (Shen) For any {b\geq 2} integer and for all {1/b < \lambda < 1}, the Mandelbrot conjecture is true, i.e.,

\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}

Remark 6 The techniques employed by Shen also allow him to show that given {\phi:\mathbb{R}\rightarrow\mathbb{R}} a {\mathbb{Z}}-periodic, non-constant, {C^2} function, and given {b\geq 2} integer, there exists {K=K(\phi,b)>1} such that

\displaystyle \textrm{dim}(\textrm{graph}(f^{\phi}_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}

for all {1/K < \lambda < 1}.

Remark 7 A previous important result towards Mandelbrot’s conjecture was obtained by Barańsky-Barány-Romanowska (in 2014): they proved that for all {b\geq 2} integer, there exists {1/b < \lambda_b < 1} such that

\displaystyle \textrm{dim}(\textrm{graph}(W_{\lambda,b})) = 2 + \frac{\log\lambda}{\log b}

for all {\lambda_b < \lambda < 1}.

The remainder of this post is dedicated to give some ideas of Shen’s proof of Theorem 1 by discussing the particular case when {1/b<\lambda<2/b} and {b\in\mathbb{N}} is large.

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It is a pleasure to announce the following two conferences (that I’m helping to organize):

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

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

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

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

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

Last time, we reduced the proof of the exponential mixing property for expanding semiflows to the following Dolgopyat-like estimate:

Proposition 1 Let {T} be an uniformly expanding Markov map on {\Delta:=(0,1)} and let {r:\Delta\rightarrow\mathbb{R}^+} be a good roof function with exponential tails.Then, there exist {\sigma_0'>0}, {T_0\geq 1}, {\beta<1} and {C>0} such that the iterates {L_s^k} of the weighted transfer operator {L_su(x):=\sum\limits_{T(y)=x} e^{-sr(y)}\frac{1}{|T'(y)|}u(y)} satisfy

\displaystyle \|L_s^k u\|_{L^2}\leq C\beta^k\left(\|u\|_{C^0}+\frac{1}{\max\{1,|t|\}}\|Du\|_{C^0}\right):= C\beta^k \|u\|_{1,t}

for all {k\in\mathbb{N}} and {s=\sigma+it} with {|\sigma|\leq\sigma_0'}, {|t|\geq T_0}.

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

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

  • (A) If {u} exhibits a high oscillation at scale {\frac{1}{|t|}} (in the sense that {\|Du\|_{C^0}\gg |t|\|u\|_{C^0}}), then we will have a “cancelation” (significant reduction of the size of {L_s^k u}) thanks to classical methods (Lasota-Yorke inequality);
  • (B) If the oscillation of {u} at scale {\frac{1}{|t|}} is not high, then we will have a “cancelation” thanks to Dolgopyat’s mechanism, i.e., a combination of high oscillations of Birkhoff sums {r^{(n)}(x)} of the roof function {r} (coming from the fact that {r} is not a {C^1}-coboundary) and the big phases {e^{-itr^{(n)}(y)}}, {|t|\geq T_0}, of the terms {e^{-sr^{(n)}(y)}} in the formula defining {L_s^n u(x)}.

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

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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.

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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.

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