Publications of Sébastien Gouëzel

Articles :

50. Random walk speed is a proper function on Teichmüller space     PDF  
A. Azemar, V. Gadre, S. Gouëzel, T. Haettel, P. Lessa and C. Uyanik
Journal of Modern Dynamics 19:815-832, 2023.

Abstract: Consider a closed surface $M$ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $M$ with finite first moment. Corresponding to each point in the Teichmüller space of $M$, there is an associated random walk on the hyperbolic plane. We show that the speed of this random walk is a proper function on the Teichmüller space of $M$, and we relate the growth of the speed to the Teichmüller distance to a basepoint. One key argument is an adaptation of Gouëzel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.

49. Minimal distance between random orbits     PDF  
S. Gouëzel, J. Rousseau and M. Stadlbauer
Preprint, 2022.

Abstract: We study the minimal distance between two orbit segments of length $n$, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called its correlation dimension (or Rényi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.

48. Exponential bounds for random walks on hyperbolic spaces without moment conditions     PDF  
S. Gouëzel
Tunisian Journal of Mathematics 4:635-671, 2022.

Abstract: We consider nonelementary random walks on general hyperbolic spaces. Without any moment condition on the walk, we show that it escapes linearly to infinity, with exponential error bounds. We even get such exponential bounds up to the rate of escape of the walk. Our proof relies on an inductive decomposition of the walk, recording times at which it could go to infinity in several independent directions, and using these times to control further backtracking.

47. Pressure at infinity and strong positive recurrence in negative curvature     PDF  
S. Gouëzel, B. Schapira and S. Tapie
Comment. Math. Helv. 98:431-508, 2023.

Abstract: In the context of geodesic flows of noncompact negatively curved manifolds, we propose three different definitions of entropy and pressure at infinity, through growth of periodic orbits, critical exponents of Poincaré series, and entropy (pressure) of invariant measures. We show that these notions coincide. Thanks to these entropy and pressure at infinity, we investigate thoroughly the notion of strong positive recurrence in this geometric context. A potential is said strongly positively recurrent when its pressure at infinity is strictly smaller than the full topological pressure. We show in particular that if a potential is strongly positively recurrent, then it admits a finite Gibbs measure. We also provide easy criteria allowing to build such strong positively recurrent potentials and many examples.

46. Classical and microlocal analysis of the X-ray transform on Anosov manifolds     PDF  
S. Gouëzel and T. Lefeuvre
Analysis & PDE 14:301-322, 2021.

Abstract: We complete the microlocal study of the geodesic X-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou in [Gui17] and pursued by Guillarmou and the second author in [GL18]. We prove new stability estimates and clarify some properties of the operator $\Pi_m$ - the generalized X-ray transform. These estimates rely on a refined version of the Livsic theorem for Anosov flows, especially on a new quantitative finite time Livsic theorem.

45. Boundary of the range of a random walk and the Fölner property     PDF  
G. Deligiannidis, S. Gouëzel and Z. Kosloff
Electronic Journal of Probability 26:110:1-39, 2021.

Abstract: The range process $R_n$ of a random walk is the collection of sites visited by the random walk up to time $n$. In this work we deal with the question of whether the range process of a random walk or the range process of a cocycle over an ergodic transformation is almost surely a Fölner sequence and show the following results: (a) The size of the inner boundary $|\partial R_n|$ of the range of recurrent aperiodic random walks on $\mathbb{Z}^2$ with finite variance and aperiodic random walks in $\mathbb{Z}$ in the standard domain of attraction of the Cauchy distribution, divided by $\frac{n}{\log^2(n)}$, converges to a constant almost surely. (b) We establish a formula for the Fölner asymptotic of transient cocycles over an ergodic probability preserving transformation and use it to show that for transient random walk on groups which are not virtually cyclic, for almost every path, the range is not a Fölner sequence. (c) For aperiodic random walks in the domain of attraction of symmetric $\alpha$- stable distributions with $1<\alpha\leq 2$, we prove a sharp polynomial upper bound for the decay at infinity of $|\partial R_n|/|R_n|$. This last result shows that the range process of these random walks is almost surely a Fölner sequence.

44. A corrected quantitative version of the Morse lemma     PDF  
S. Gouëzel and V. Shchur
Journal of Functional Analysis 277:1248-1258, 2019.

Abstract: There is a gap in the proof of the main theorem in the article [Sh13] on optimal bounds for the Morse lemma in Gromov-hyperbolic spaces. We correct this gap, showing that the main theorem of [Sh13] is correct. We also describe a computer certification of this result

43. Ruelle spectrum of linear pseudo-Anosov maps     PDF  
F. Faure, S. Gouëzel and E. Lanneau
Journal de l'École Polytechnique 6:811-877, 2019.

Abstract: The Ruelle resonances of a dynamical system are spectral data describing the precise asymptotics of correlations. We classify them completely for a class of chaotic two-dimensional maps, the linear pseudo-Anosov maps, in terms of the action of the map on cohomology. As applications, we obtain a full description of the distributions which are invariant under the linear flow in the stable direction of such a linear pseudo-Anosov map, and we solve the cohomological equation for this flow.

42. Asymptotic combinatorics of Artin-Tits monoids and of some other monoids     PDF  
S. Abbes, S. Gouëzel, V. Jugé and J. Mairesse
Journal of Algebra 525:497-561, 2019.

Abstract: We introduce methods to study the combinatorics of the normal form of large random elements in Artin-Tits monoids. These methods also apply in an axiomatic framework that encompasses other monoids such as dual braid monoids.

41. Growth of normalizing sequences in limit theorems for conservative maps     PDF  
S. Gouëzel
Electronic Communications in Probability 23-99, 2018.

Abstract: We consider normalizing sequences that can give rise to nondegenerate limit theorems for Birkhoff sums under the iteration of a conservative map. Most classical limit theorems involve normalizing sequences that are polynomial, possibly with an additional slowly varying factor. We show that, in general, there can be no nondegenerate limit theorem with a normalizing sequence that grows exponentially, but that there are examples where it grows like a stretched exponential, with an exponent arbitrarily close to $1$.

40. Variations around Eagleson's Theorem on mixing limit theorems for dynamical systems     PDF  
S. Gouëzel
Ergodic Theory and Dynamical Systems 40, 3368-3374, 2020.

Abstract: Eagleson's Theorem asserts that, given a probability-preserving map, if renormalized Birkhoff sums of a function converge in distribution, then they also converge with respect to any probability measure which is absolutely continuous with respect to the invariant one. We prove a version of this result for almost sure limit theorems, extending results of Korepanov. We also prove a version of this result, in mixing systems, when one imposes a conditioning both at time $0$ and at time $n$.

39. Quantitative Pesin theory for Anosov diffeomorphisms and flows     PDF  
S. Gouëzel and L. Stoyanov
Ergodic Theory and Dynamical Systems 39:159-200, 2019.

Abstract: Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure preserving dynamical system is explicitly controlled. In uniformly hyperbolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows, and counterexamples illustrating the necessity of suitable conditions on the cocycle.

38. Uniform measures on braid monoids and dual braid monoids     PDF  
S. Abbes, S. Gouëzel, V. Jugé and J. Mairesse
Journal of Algebra 473:627-666, 2017.

Abstract: We aim at studying the asymptotic properties of typical positive braids, respectively positive dual braids. Denoting by $\mu_k$ the uniform distribution on positive (dual) braids of length $k$, we prove that the sequence $(\mu_k)_k$ converges to a unique probability measure $\mu_{\infty}$ on infinite positive (dual) braids. The key point is that the limiting measure $\mu_{\infty}$ has a Markovian structure which can be described explicitly using the combinatorial properties of braids encapsulated in the Möbius polynomial. As a by-product, we settle a conjecture by Gebhardt and Tawn (J. Algebra, 2014) on the shape of the Garside normal form of large uniform braids.

37. Large and moderate deviations for bounded functions of slowly mixing Markov chains     PDF  
J. Dedecker, S. Gouëzel and F. Merlevède
Stochastics and Dynamics 18:1850017, 2018.

Abstract: We consider Markov chains which are polynomially mixing, in a weak sense expressed in terms of the space of functions on which the mixing speed is controlled. In this context, we prove polynomial large and moderate deviations inequalities. These inequalities can be applied in various natural situations coming from probability theory or dynamical systems. Finally, we discuss examples from these various settings showing that our inequalities are sharp.

36. Subadditive and multiplicative ergodic theorems     PDF
S. Gouëzel and A. Karlsson
Journal of the EMS 22:1893-1915, 2020.

Abstract: A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman's subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of Karlsson-Ledrappier, showing that the growth of a random product of semi-contractions is always directed by some horofunction. We discuss applications of this result to ergodic cocycles of bounded linear operators, holomorphic maps and topical operators, as well as a random mean ergodic theorem.

35. Analyticity of the entropy and the escape rate of random walks in hyperbolic groups     PDF  
S. Gouëzel
Discrete Analysis 2017:7, 1-37.

Abstract: We consider random walks on a non-elementary hyperbolic group endowed with a word distance. To a probability measure on the group are associated two numerical quantities, the rate of escape and the entropy. On the set of admissible probability measures whose support is contained in a given finite set, we show that both quantities depend in an analytic way on the probability measure. Our spectral techniques also give a new proof of the central limit theorem, and imply that the corresponding variance is analytic.

34. Entropy and drift in word hyperbolic groups     PDF  
S. Gouëzel, F. Mathéus and F. Maucourant
Inventiones Mathematicae 211:1201-1255, 2018.

Abstract: The fundamental inequality of Guivarc'h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary hyperbolic group which is not virtually free, endowed with a word distance, the fundamental inequality is strict for symmetric measures with finite support, uniformly for measures with a given support. This answers a conjecture of S. Lalley. For admissible measures, this is proved using previous results of Ancona and Blachère-Haïssinsky-Mathieu. For non-admissible measures, this follows from a counting result, interesting in its own right: we show that, in any infinite index subgroup, the number of non-distorted points is exponentially small. The uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup.

33. Subgaussian concentration inequalities for geometrically ergodic Markov chains     PDF  
S. Gouëzel and J. Dedecker
Electronic Communications in Probability 20:1-12, 2015.

Abstract: We prove that an irreducible aperiodic Markov chain is geometrically ergodic if and only if any separately bounded functional of the stationary chain satisfies an appropriate subgaussian deviation inequality from its mean.

32. Moment bounds and concentration inequalities for slowly mixing dynamical systems     PDF  
S. Gouëzel and I. Melbourne
Electronic Journal of Probability 93: 1-30, 2014.

Abstract: We obtain optimal moment bounds for Birkhoff sums, and optimal concentration inequalities, for a large class of slowly mixing dynamical systems, including those that admit anomalous diffusion in the form of a stable law or a central limit theorem with nonstandard scaling $(n\log n)^{1/2}$.

31. A numerical lower bound for the spectral radius of random walks on surface groups     PDF  
S. Gouëzel
Combinatorics, Probability and Computing, 24: 838-856, 2015.

Abstract: Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus $2$ surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi.

Additional comments: This file contains additionally the implementation of the algorithms described in the paper, in its appendix.

30. Martin boundary of random walks with unbounded jumps in hyperbolic groups     PDF  
S. Gouëzel
Annals of Probability 43:2374-2404, 2015.

Abstract: Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is known since the work of Ancona and Gouëzel-Lalley that the Martin boundary coincides with the geometric boundary. The goal of this paper is to weaken the finite support assumption. We first show that, in any non-amenable group, there exist probability measures with exponential tails giving rise to pathological Martin boundaries. Then, for probability measures with superexponential tails in hyperbolic groups, we show that the Martin boundary coincides with the geometric boundary by extending Ancona's inequalities. We also deduce asymptotics of transition probabilities for symmetric measures with superexponential tails.

29. Sharp lower bounds for the asymptotic entropy of symmetric random walks     PDF  
S. Gouëzel, F. Mathéus and F. Maucourant
Groups, Geometry, and Dynamics 9:711-735, 2015.

Abstract: The entropy, the spectral radius and the drift are important numerical quantities associated to random walks on countable groups. We prove sharp inequalities relating those quantities for walks with a finite second moment, improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also deduce inequalities between these quantities and the volume growth of the group. Finally, we show that the equality case in our inequality is rather rigid.

28. Local limit theorem for symmetric random walks in Gromov-hyperbolic groups     PDF  
S. Gouëzel
Journal of the A.M.S.27:893-928, 2014.

Abstract: Completing a strategy of Gouëzel and Lalley, we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non-elementary Gromov-hyperbolic group: denoting by $R$ the inverse of the spectral radius of the random walk, the probability to return to the identity at time $n$ behaves like $C R^{-n}n^{-3/2}$. An important step in the proof is to extend Ancona's results on the Martin boundary up to the spectral radius: we show that the Martin boundary for $R$-harmonic functions coincides with the geometric boundary of the group. In an appendix, we explain how the symmetry assumption of the measure can be dispensed with for surface groups.

Additional comments: The finiteness of the support has been relaxed in "Martin boundary of random walks with with unbounded jumps in hyperbolic groups".

27. Optimal concentration inequalities for dynamical systems     PDF  
J.-R. Chazottes and S. Gouëzel
Communications in Mathematical Physics 316:843-889, 2012.

Abstract: For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of $n$ variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables.

Additional comments: In fact, the inequalities we obtain in this paper are not as optimal as we thought. They have been improved (and extended to systems that mix more slowly) in "Moment bounds and concentration inequalities for slowly mixing dynamical systems".

26. Correlation asymptotics from large deviations in dynamical systems with infinite measure     PDF  
S. Gouëzel
Colloquium Mathematicum 125:193-212, 2011.

Abstract: We extend a result of Doney on renewal sequences with infinite mean to renewal sequences of operators. As a consequence, we get precise asymptotics for the transfer operator and for correlations in dynamical systems preserving an infinite measure (including intermittent maps with an arbitrarily neutral fixed point).

25. Random walks on co-compact Fuchsian groups     PDF  
S. Gouëzel and S. Lalley
Annales Scientifiques de l'ENS 46:129-173, 2013.

Abstract: It is proved that the Green's function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence $R$. It is also shown that Ancona's inequalities extend to $R$, and therefore that the Martin boundary for $R$-potentials coincides with the natural geometric boundary $S^{1}$, and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, $p^n(x,y)\sim C_{x,y}R^{-n}n^{-3/2}$.

Additional comments: The planarity assumption has been removed in the subsequent paper "Local limit theorem for symmetric random walks in Gromov-hyperbolic groups", and the finiteness of the support has been relaxed in "Martin boundary of measures with infinite support in hyperbolic groups".

24. The almost sure invariance principle for unbounded functions of expanding maps     PDF  
J. Dedecker, S. Gouëzel and F. Merlevède
ALEA 9:141-163, 2012.

Abstract: We consider two classes of piecewise expanding maps $T$ of $[0,1]$: a class of uniformly expanding maps for which the Perron-Frobenius operator has a spectral gap in the space of bounded variation functions, and a class of expanding maps with a neutral fixed point at zero. In both cases, we give a large class of unbounded functions $f$ for which the partial sums of $f\circ T^i$ satisfy an almost sure invariance principle. This class contains piecewise monotonic functions (with a finite number of branches) such that: For uniformly expanding maps, they are square integrable with respect to the absolutely continuous invariant probability measure. For maps having a neutral fixed point at zero, they satisfy an (optimal) tail condition with respect to the absolutely continuous invariant probability measure.

23. Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow     PDF  
A. Avila and S. Gouëzel
Annals of Mathematics 178:385-442, 2013.

Abstract: We consider the $\mathrm{SL}(2,\mathbb{R})$ action on moduli spaces of quadratic differentials. If $\mu$ is an $\mathrm{SL}(2,\mathbb{R})$-invariant probability measure, crucial information about the associated representation on $L^2(\mu)$ (and in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded in the part of the spectrum of the corresponding foliated hyperbolic Laplacian that lies in $(0,1/4)$ (which controls the contribution of the complementary series). Here we prove that the essential spectrum of an invariant algebraic measure is contained in $[1/4,\infty)$, i.e., for every $\delta>0$, there are only finitely many eigenvalues (counted with multiplicity) in $(0,1/4-\delta)$. In particular, all algebraic invariant measures have a spectral gap.

22. Banach spaces for piecewise cone hyperbolic maps     PDF  
V. Baladi and S. Gouëzel
Journal of Modern Dynamics 4:91-137, 2010.

Abstract: We consider piecewise cone hyperbolic systems satisfying a bunching condition and we obtain a bound on the essential spectral radius of the associated weighted transfer operators acting on anisotropic Sobolev spaces. The bunching condition is always satisfied in dimension two, and our results give a unifying treatment of the work of Demers-Liverani and our previous work. When the complexity is subexponential, our bound implies a spectral gap for the transfer operator corresponding to the physical measures in many cases (for example if $T$ preserves volume, or if the stable-dimension is equal to $1$ and the unstable dimension is not zero).

21. Almost sure invariance principle for dynamical systems by spectral methods     PDF  
S. Gouëzel
Annals of Probability 38:1639-1671, 2010.

Abstract: We prove the almost sure invariance principle for stationary $\mathbb{R}^d$-valued processes (with dimension-independent very precise error terms), solely under a strong assumption on the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains, using strong or weak spectral perturbation arguments.

20. Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains     PDF  
J. Dedecker, S. Gouëzel and F. Merlevède
Annales de l'IHP Probabilités et Statistiques 46:796-821, 2010.

Abstract: We consider a large class of piecewise expanding maps $T$ of $[0,1]$ with a neutral fixed point, and their associated Markov chain $Y_i$ whose transition kernel is the Perron-Frobenius operator of $T$ with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions $f$ for which the partial sums of $f\circ T^i$ satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of $f(Y_i)$ satisfy a strong invariance principle. When the class is larger, so that the partial sums of $f\circ T^i$ may belong to the domain of normal attraction of a stable law of index $p\in (1, 2)$, we show that the almost sure rates of convergence in the strong law of large numbers are the same as in the corresponding i.i.d. case.

Additional comments: There is a mistake in the proof of Theorem 1.7, in the case $p=2$. On Page 819, the first inequality in the second equation after (4.13) is not correct: to handle the second term coming from the previous equation, one has a series $\sum_{n>A} \frac{1}{n (\ln n)^{2b}}$, whose asymptotics is not $1/(\ln A)^{2b}$... When $p<2$, on the contrary, one has regularly varying functions of index different from -1, so everything is OK. The proof can easily be corrected, instead of using the (too naive) maximal inequality from [2], one should use Proposition 1.11.

19. Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps     PDF  
S. Gouëzel
Israel Journal of Mathematics 180:1-41, 2010.

Abstract: We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For $L^2$ observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.

18. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions     PDF  
S. Gouëzel
Discrete and Continuous Dynamical Systems 24:1205-1208, 2009.

Abstract: We construct a uniformly expanding map of the interval, preserving Lebesgue measure, such that the corresponding transfer operator admits a spectral gap on the space of Lipschitz functions, but does not act continuously on the space of bounded variation functions.

17. Good Banach spaces for piecewise hyperbolic maps via interpolation     PDF  
V. Baladi and S. Gouëzel
Annales de l'IHP Analyse non linéaire 26:1453-1481, 2009.

Abstract: We introduce a weak transversality condition for piecewise $C^{1+\alpha}$ and piecewise hyperbolic maps which admit a $C^{1+\alpha}$ stable distribution. We show bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkin type which are better than previously known estimates (when our assumption on the stable distribution holds). In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) but gives a new result on bounded multipliers. Our method applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results on a simpler scale of Banach spaces.

16. A Borel-Cantelli lemma for intermittent interval maps     PDF  
S. Gouëzel
Nonlinearity 20:1491-1497, 2007.

Abstract: We consider intermittent maps $T$ of the interval, with an absolutely continuous invariant probability measure $\mu$. Kim showed that there exists a sequence of intervals $A_n$ such that $\sum \mu(A_n)=\infty$, but $\{A_n\}$ does not satisfy the dynamical Borel-Cantelli lemma, i.e., for almost every $x$, the set $\{n : T^n(x)\in A_n\}$ is finite. If $\sum Leb(A_n)=\infty$, we prove that $\{A_n\}$ satisfies the Borel-Cantelli lemma. Our results apply in particular to some maps $T$ whose correlations are not summable.

15. Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences    PDF  
S. Gouëzel
Duke Mathematical Journal 147:192-284, 2009.

Abstract: We study skew-products of the form $(x,\omega)\mapsto (Tx, \omega+\phi(x))$ where $T$ is a nonuniformly expanding map on a space $X$, preserving a (possibly singular) probability measure $\tilde\mu$, and $\phi:X\to S^1$ is a $C^1$ function. Under mild assumptions on $\tilde\mu$ and $\phi$, we prove that such a map is exponentially mixing, and satisfies the central and local limit theorems. These results apply to a random walk related to the Farey sequence, thereby answering a question of Guivarc'h and Raugi.

Additional comments: To apply the main results of the article to Farey sequences, one has at some point (in the proof of Lemma 6.4) to check that an explicit quantity is nonzero. I have made the computation with Maple, the output is available here.

14. Limit theorems for coupled interval maps    PDF  
J.-B. Bardet, S. Gouëzel and G. Keller
Stochastics and Dynamics 7:17-36, 2007.

Abstract: We prove a local limit theorem for Lipschitz continuous observables on a weakly coupled lattice of piecewise expanding interval maps. The core of the paper is a proof that the spectral radii of the Fourier-transfer operators for such a system are strictly less than $1$. This extends the approach of [KL2] where the ordinary transfer operator was studied.

13. Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties    PDF  
S. Gouëzel and C. Liverani
Journal of Differential Geometry 79:433-477, 2008.

Abstract: Compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set. This provides a self-contained theory that not only reproduces all the known classical results but gives also new insights on the statistical properties of these systems.

12. On almost-sure versions of classical limit theorems for dynamical systems     PDF  
J.-R. Chazottes and S. Gouëzel
Probability Theory and Related Fields 138:195-234, 2007.

Abstract: The purpose of this article is to support the idea that ``whenever we can prove a limit theorem in the classical sense for a dynamical system, we can prove a suitable almost-sure version based on an empirical measure with log-average''. We follow three different approaches: martingale methods, spectral methods and induction arguments. Our results apply, among others, to Axiom A maps or flows, to systems inducing a Gibbs-Markov map, and to the stadium billiard.

Along the way, we also mention the notion of "convergence with tight maxima", which makes it possible to induce standard limit theorems without any assumption on the return time.

11. Exponential mixing for the Teichmüller flow     PDF
A. Avila, S. Gouëzel and J.-C. Yoccoz
Publications mathématiques de l'IHES 104:143-211, 2006.

Abstract: We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Höder observables. A geometric consequence is that the $\mathrm{SL}(2,\mathbb{R})$ action in the moduli space has a spectral gap.

The proof involves a fine combinatorial study of interval exchange maps and Rauzy diagram to get exponential return estimates to a good set, and then a spectral analysis of transfer operators following Dolgopyat's arguments for Anosov flows.

10. Smoothness of solenoidal attractors     PDF
A. Avila, S. Gouëzel and M. Tsujii
Discrete and Continuous Dynamical Systems 15:21-35, 2006.

Abstract: We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$: \[ T(x,y)=(\ell x, \lambda y+f(x)) \] where $\ell\ge 2$, $0<\lambda<1$ and $f$ is a $C^r$ function on $S^1$. We show that, if $\lambda^{1+2s}\ell>1$ for some $0\leq s< r-2$, the density of the SBR measure for $T$ is contained in the Sobolev space $W^s(S^1\times \mathbb{R})$ for almost all ($C^r$ generic, at least) $f$.

Additional comments: The proof is spectral theoretic, and consists in constructing good Banach spaces on which the transfer operator has a spectral gap. These spaces are built by mixing the ideas of "Banach spaces adapted to Anosov systems" and Fourier arguments coming from Tsujii's transversality ideas. It should be noted that, using the spaces of V. Baladi and M. Tsujii in "Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms", one gets a more efficient and transparent argument (see the preprint "Decay of correlations in suspension semi-flows of angle-multiplying maps" by M. Tsujii).

9. Limit theorems in the stadium billiard    PDF
P. Bálint and S. Gouëzel
Communications in Mathematical Physics 263:451-512, 2006

Abstract: We prove that the Birkhoff sums for ``almost every'' relevant observable in the stadium billiard obey a non-standard limit law. More precisely, the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a $\sqrt{n\log n}$ normalization is needed. As one of the two key steps in the argument, we obtain a limit theorem that holds in Young towers with exponential return time statistics in general, an abstract result that seems to be applicable to many other situations.

8. Regularity of coboundaries for non uniformly expanding Markov maps     PDF
S. Gouëzel
Proceedings of the American Mathematical Society 134:391-401, 2006

Abstract: We prove that solutions $u$ of the equation $f=u-u\circ T$ are automatically Hölder continuous when $f$ is Hölder continuous, and $T$ is non uniformly expanding and Markov. This result applies in particular to Young towers and to intermittent maps.

7. Banach spaces adapted to Anosov systems     PDF
S. Gouëzel and C. Liverani
Ergodic Theory and Dynamical Systems 26:189-217, 2006

Abstract: We study the spectral properties of the Ruelle-Perron-Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the $C^\infty$ case, the essential spectral radius is arbitrarily small, which yields a description of the correlations with arbitrary precision. Moreover, we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the SRB measure, the variance for the CLT, the rates of decay for smooth observable, etc.).

Additional comments: The arguments given in this paper are definitely not optimal. Indeed, it is possible to construct spaces on which the limitation of our approach - the fact that the parameter $p$ has to be an integer - is removed, as shown by V. Baladi and M. Tsujii in "Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms". Moreover, our proofs are too complicated even in our setting, and our spaces can be given a more geometric interpretation (which makes further extensions possible). So, my advice to the reader is to skip this paper and read directly the aforementionned paper by Baladi and Tsujii, or "Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties" by Liverani and myself. The result on weak perturbation theory with several derivatives has been precised and refined in "Necessary and sufficient conditions for limit theorems in Gibbs-Markov maps". Note that there is a small gap at the end of the proof, where we use implicitly that $(z-\mathcal{L}_0)^{-1}$ acts boundedly on $\mathcal{B}_0$ while this is not true in general. This gap is fixed in "Necessary and sufficient conditions for limit theorems in Gibbs-Markov maps".

6. Decay of correlations for nonuniformly expanding systems     PDF
S. Gouëzel
Bulletin de la Société Mathématique de France 134:1-31, 2006

Abstract: We estimate the speed of decay of correlations for general nonuniformly expanding dynamical systems, using estimates on the time the system takes to become really expanding. Our method can deal with fast decays, such as exponential or stretched exponential. We prove in particular that the correlations of the Alves-Viana map decay in $O(e^{-c \sqrt{n}})$.

Additional comments: A very interesting paper by V. Pinheiro ("Sinai-Ruelle-Bowen measures for weakly expanding maps", Nonlinearity 19 (2006), 1185-1200) gave me some ideas to improve slightly on the results of my paper (and especially to remove the extra factor $\log n$ in Theorem 1.3, or to work using only estimates on the first hyperbolic time). These results are presented in the following informal note.

5. Statistical properties of a skew product with a curve of neutral points     PDF
S. Gouëzel
Ergodic Theory and Dynamical Systems 27:123-151, 2007

Abstract: We study a skew product with a curve of neutral points. We show that there exists a unique absolutely continuous invariant probability measure, and that the Birkhoff averages of a sufficiently smooth observable converge to a normal law or a stable law, depending on the average of the observable along the neutral curve.

Additional comments

4. Berry-Esseen theorem and local limit theorem for non uniformly expanding maps     PDF
S. Gouëzel
Annales de l'IHP Probabilités et Statistiques 41:997-1024, 2005

Abstract: In Young towers with sufficiently small tails, the Birkhoff sums of Hölder continuous functions satisfy a central limit theorem with speed $O(1/\sqrt{n})$, and a local limit theorem. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.

3. Central limit theorem and stable laws for intermittent maps     PDF
S. Gouëzel
Probability Theory and Related Fields 128:82-122, 2004

Abstract: In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at $0$ of the form $x+x^{1+\alpha}$, for $\alpha\in (0,1)$. In particular, for $\alpha>1/2$, we show that the Birkhoff sums of a Hölder observable $f$ converge to a normal law or a stable law, depending on whether $f(0)=0$ or $f(0)\not=0$. The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in non-commutative Banach algebras.

Additional comments: The spectral method used in this article to prove the limit theorems is definitely not the good one: the elementary arguments used by Zweimüller or Melbourne and Török are at the same time more elementary and more efficient to do that (see the article "On almost-sure versions of classical limit theorems for dynamical systems" for a version of this argument, and "stable laws for the doubling map" for a more or less final version of this argument). However, the great interest of this spectral method is that it can be extended to prove finer limit theorems which are not accessible to elementary methods, such as the local limit theorem (see "Berry-Esseen theorem and local limit theorem for non uniformly expanding maps").

There are also small mistakes (particularly concerning the computation of explicit constants) in the published paper (the last lines of Page 85, and Item 2 on Page 88), that are corrected in the preprint version here.

2. Sharp polynomial estimates for the decay of correlations     PDF
S. Gouëzel
Israel Journal of Mathematics 139:29-65, 2004

Abstract: We generalize a method developed by Sarig to obtain polynomial lower bounds for correlation functions for maps with a countable Markov partition. A consequence is that LS Young's estimates on towers are always optimal. Moreover, we show that, for functions with zero average, the decay rate is better, gaining a factor $1/n$. This implies a Central Limit Theorem in contexts where it was not expected, e.g. $x+Cx^{1+\alpha}$ with $1/2\leq \alpha<1$. The method is based on a general result on renewal sequences of operator, and gives an asymptotic estimate up to any precision of such operators.

Additional comments: The method described in the article is very well suited to speeds of decay of the form $1/n^\alpha$. However, for more exotic speeds such as $1/\log n$ or $e^{-\sqrt{n}}$, the "elementary" part of the article, using Hölder regularity, should be replaced by a purely Banach algebra argument. This is done in full generality in chapter 2 of my (unpublished) thesis.

1. Spectre de l'opérateur de transfert en dimension 1     PDF
S. Gouëzel
Manuscripta Mathematica 106:365-403, 2001

Abstract: On étudie les propriétés spectrales d'un opérateur de transfert $M\Phi(x)=\sum_{\omega}g_{\omega}(x)\Phi(\psi_\omega x)$ agissant sur les fonctions à variation bornée. Après des estimations - utilisant une intégrale symétrique - du rayon spectral et du rayon spectral essentiel, on considère le déterminant dynamique $Det^\#(Id+zM)$. On généralise au cas des poids discontinus le résultat de Baladi-Ruelle (pour des poids continus) sur le lien entre zéros du déterminant dynamique et spectre de l'opérateur de transfert. La preuve, par régularisation des poids, utilise un résultat spectral assurant la surjectivité automatique de certaines applications entre sous-espaces caractéristiques d'opérateurs.

Other documents :

9. Subadditive cocycles and horofunctions     PDF
S. Gouëzel
Talk given at ICM 2018

Abstract : The aim of this text is to present and put in perspective the results we have proved with Anders Karlsson. The topic of this article is the study, in an ergodic theoretic context, of some subadditivity properties, and their relationships with dynamical questions with a more geometric flavor, dealing with the asymptotic behavior of random semicontractions on general metric spaces. This text is translated from an article written in French on the occasion of the first congress of the French Mathematical Society. The proof of the main ergodic-theoretic result has been completely formalized and checked in the computer proof assistant Isabelle/HOL.

8. Méthodes entropiques pour les convolutions de Bernoulli, d'après Hochman, Shmerkin, Breuillard, Varjú     PDF
S. Gouëzel
Exposé au séminaire Bourbaki 2018

Résumé : La convolution de Bernoulli de paramètre $\lambda \in ]1/2, 1[$ est la loi de $\sum \xi_n \lambda^n$, où les $\xi_n$ forment une suite de variables de Bernoulli non biaisées. On conjecture depuis les travaux fondateurs d'Erdös et Kahane que cette mesure réelle est absolument continue par rapport à la mesure de Lebesgue lorsque $\lambda$ n'est pas l'inverse d'un nombre de Pisot. Cette question, malgré son apparente simplicité, est extrêmement délicate et encore ouverte. Elle a motivé au fil du temps le développement de différentes techniques qui ont ensuite pu être appliquées dans des contextes beaucoup plus généraux. Cet exposé sera consacré à la méthode entropique, introduite récemment par Hochman, qui fait le lien avec le monde de la combinatoire additive et a permis des développements spectaculaires.

7. Cocycles sous-additifs et horofonctions     PDF
S. Gouëzel
Séminaires et Congrès 31:19-38, 2017

Résumé : L'objectif de ce texte, correspondant à l'exposé que j'ai donné à l'occasion du premier congrès de la SMF à Tours, est de présenter et de mettre en perspective les résultats obtenus avec Anders Karlsson dans l'article [Gouezel-Karlsson]. Le thème général de cet article est l'étude dans un contexte de théorie ergodique de certaines propriétés de sous-additivité, et leur lien avec des questions de dynamique nettement plus géométriques concernant le comportement asymptotique de semi-contractions aléatoires sur des espaces métriques généraux.

6. Spectre du flot géodésique en courbure négative, d'après F. Faure et M. Tsujii     PDF
S. Gouëzel
Astérisque 380:325-353, 2016, Séminaire Bourbaki. Vol. 2014/2015 (2016), exposé 1098

Résumé : Étant donnée une variété compacte à courbure négative, on peut combiner les longueurs de ses géodésiques fermées pour former une fonction zeta naturelle. En courbure constante, ses zéros non triviaux se trouvent sur des droites verticales explicites d'après les travaux de Selberg (1956). Faure et Tsujii ont établi une généralisation remarquable et inattendue de ce résultat en courbure variable : les zéros se situent en général dans des bandes verticales et, pour une fonction zeta spécifique due à Gutzwiller-Voros, les zéros de la première bande s'alignent asymptotiquement sur une droite verticale. Les motivations et les techniques relèvent à la fois des systèmes dynamiques et de l'analyse semi-classique.

5. Limit theorems in dynamical systems using the spectral method     PDF
S. Gouëzel
Proceedings of Symposia in Pure Mathematics 89:161-193, 2015

Abstract: There are numerous techniques to prove probabilistic limit theorems for dynamical systems. These notes are devoted to one of these methods, the Nagaev-Guivarc'h spectral method, which extends to dynamical systems the usual proof of the central limit theorem relying on characteristic functions. We start with the the simplest example (expanding maps of the interval), where everything is elementary. We then consider more recent (and more involved) applications of this method, on the one hand to get the convergence to stable laws in intermittent maps, on the other hand to obtain precise results on the almost sure approximation by a Brownian motion.

4. Mon habilitation à diriger des recherches Comportement quantitatif de certains systèmes dynamiques. Exemples et applications     PDF

3. Stable laws for the doubling map     PDF
S. Gouëzel

Abstract: We prove stable limit theorems for the functions $f_\alpha(x)=x^{-\alpha}$, $\alpha\geq 1/2$, under the iteration of the doubling map $T:x\to 2x \mod 1$. The limiting distributions are smaller (resp. larger) than the sum of corresponding i.i.d. random variables when $\alpha>1$ (resp. $<1$).

2. Un théorème de Kerckhoff, Masur et Smillie : Unique ergodicité sur les surfaces plates     PDF
S. Gouëzel and E. Lanneau
Séminaires et Congrès 20:113-145, 2010

Résumé : Ces notes correspondent à un cours donné lors de l'école thématique de théorie ergodique au C.I.R.M. à Marseille en avril 2006. Nous présentons et démontrons un théorème de Kerckhoff, Masur et Smillie sur l'unique ergodicité du flot directionnel sur une surface de translation dans presque toutes les directions. La preuve suit essentiellement celle présentée dans un survol de Masur et Tabachnikov. Nous donnons une preuve complète et élémentaire du théorème.

1. My PhD thesis Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes     PDF

Proof assistants :

6. A formalization of the change of variables formula for integrals in mathlib     pdf
S. Gouëzel
CICM 2022, Lecture Notes in Computer Science 13467:3-18, 2022

Abstract : We report on a formalization of the change of variables formula in integrals, in the mathlib library for Lean. Our version of this theorem is extremely general, and builds on developments in linear algebra, analysis, measure theory and descriptive set theory. The interplay between these domains is transparent thanks to the highly integrated development model of mathlib.

5. Formalizing the Gromov-Hausdorff distance     pdf
S. Gouëzel
CICM-WS 2021, CEUR-WS vol. 3377

Abstract : The Gromov-Hausdorff space is usually defined in textbooks as ``the space of all compact metric spaces up to isometry''. We describe a formalization of this notion in the Lean proof assistant, insisting on how we need to depart from the usual informal viewpoint of mathematicians on this object to get a rigorous formalization.

4. the Gromov-Hausdorff distance     Link
S. Gouëzel
Lean 3 2019.

Abstract : The Gromov-Hausdorff distance on the space of nonempty compact metric spaces up to isometry.

We introduces the space of all nonempty compact metric spaces, up to isometry, and endow it with a metric space structure. The distance, known as the Gromov-Hausdorff distance, is defined as follows: given two nonempty compact spaces $X$ and $Y$, their distance is the minimum Hausdorff distance between all possible isometric embeddings of X and Y in all metric spaces. To define properly the Gromov-Hausdorff space, we consider the non-empty compact subsets of $\ell^\infty(\mathbb{R})$ up to isometry, which is a well-defined type, and define the distance as the infimum of the Hausdorff distance over all embeddings in $\ell^\infty(\mathbb{R})$. We prove that this coincides with the previous description, as all separable metric spaces embed isometrically into $\ell^\infty(\mathbb{R})$, through an embedding called the Kuratowski embedding. To prove that we have a distance, we should show that if spaces can be coupled to be arbitrarily close, then they are isometric. More generally, the Gromov-Hausdorff distance is realized, i.e., there is a coupling for which the Hausdorff distance is exactly the Gromov-Hausdorff distance. This follows from a compactness argument, essentially following from Arzela-Ascoli.

We prove the most important properties of the Gromov-Hausdorff space: it is a polish space, i.e., it is complete and second countable. We also prove the Gromov compactness criterion.

3. Gromov Hyperbolicity     Link
S. Gouëzel
Isabelle/HOL 2018

Abstract : A geodesic metric space is Gromov hyperbolic if all its geodesic triangles are thin, i.e., every side is contained in a fixed thickening of the two other sides. While this definition looks innocuous, it has proved extremely important and versatile in modern geometry since its introduction by Gromov. We formalize the basic classical properties of Gromov hyperbolic spaces, notably the Morse lemma asserting that quasigeodesics are close to geodesics, the invariance of hyperbolicity under quasi-isometries, we define and study the Gromov boundary and its associated distance, and prove that a quasi-isometry between Gromov hyperbolic spaces extends to a homeomorphism of the boundaries. We also classify the isometries of hyperbolic spaces into elliptic, parabolic and loxodromic ones, both in terms of translation length and of fixed points at infinity. We also prove a less classical theorem, by Bonk and Schramm, asserting that a Gromov hyperbolic space embeds isometrically in a geodesic Gromov-hyperbolic space. As the original proof uses a transfinite sequence of Cauchy completions, this is an interesting formalization exercise. Along the way, we introduce basic material on isometries, quasi-isometries, geodesic spaces, the Hausdorff distance, the Cauchy completion of a metric space, and the exponential on extended real numbers.

2. $L^p$ spaces     Link
S. Gouëzel
Isabelle/HOL 2016

Abstract : $L^p$ is the space of functions whose p-th power is integrable. It is one of the most fundamental Banach spaces that is used in analysis and probability. We develop a framework for function spaces, and then implement the Lp spaces in this framework using the existing integration theory in Isabelle/HOL. Our development contains most fundamental properties of $L^p$ spaces, notably the Hölder and Minkowski inequalities, completeness of $L^p$, duality, stability under almost sure convergence, multiplication of functions in $L^p$ and $L^q$, stability under conditional expectation.

1. Ergodic theory     Link
S. Gouëzel
Isabelle/HOL 2016

Abstract : Ergodic theory is the branch of mathematics that studies the behaviour of measure preserving transformations, in finite or infinite measure. It interacts both with probability theory (mainly through measure theory) and with geometry as a lot of interesting examples are from geometric origin. We implement the first definitions and theorems of ergodic theory, including notably Poincaré recurrence theorem for finite measure preserving systems (together with the notion of conservativity in general), induced maps, Kac's theorem, Birkhoff theorem (arguably the most important theorem in ergodic theory), and variations around it such as conservativity of the corresponding skew product, or Atkinson lemma, and Kingman theorem. Using this material, we formalize completely the proof of the main theorems of [Gouëzel-Karlsson] and [Gouëzel].

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