Sébastien Gouëzel

Banach spaces adapted to Anosov systems

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 ∞ case, the essential spectral radius is arbitrarily small, which yieldsa descriptionof the correlationswith arbitraryprecision. 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 Sinai-Ruelle-Bowenmeasure, the variancefor the centrallimit theorem, the rates of decay for smooth observable, etc.).

Sharp polynomial estimates for the decay of correlations

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+Cx1+α with 1/2⩽α<1. The method is based on a general result on renewal sequences of operators, and gives an asymptotic estimate up to any precision of such operators.

Almost sure invariance principle for dynamical systems by spectral methods.

The almost sure invariance principle is a very strong reinforcement of the central limit theorem: it ensures that the trajectories of a process can be matched with the trajectories of a brownian motion in such a way that, almost surely, the error between the trajectories is negligible compared to the size of the trajectory (the result can be more or less precise, depending on the specific error term one can obtain). This kind of results has a lot of consequences, see e.g. [MN09] and references therein. Such results are well known for one-dimensional processes, either independent or weakly dependent (see among many others [DP84, HK82]), and for independent higher dimensional processes [Ein89, Zăı98]. However, for weakly dependent higher dimensional processes, difficulties arise since the techniques relying on Skorokhod representation theorem do not work efficiently. In this direction, an approximation argument introduced by [BP79] was recently generalized to a large class of weakly dependent sequences in [MN09]: their results give explicit error terms in the vector–valued almost sure invariance principle, and are applicable when the variables under study can be well approximated with respect to a suitably chosen filtration. In particular, these results apply to a large range of dynamical systems when they have some markovian behavior and sufficient hyperbolicity. Unfortunately, it is quite common to encounter dynamical systems for which there is no natural well-behaved filtration. It is nevertheless often easy to prove classical limit theorems, by using another class of arguments relying on spectral theory: these arguments automatically yield a very precise description of the characteristic functions of the process under study, thereby implying limit theorems. It is therefore desirable to develop an abstract argument, showing that enough control on the characteristic functions of a process implies the almost sure invariance principle, for vector–valued observables. This is our goal in this paper.We prove the almost sure invariance principle for stationary ℝ d -valued random processes (with very precise dimension-independent error terms), solely under a strong assumption concerning 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.

Statistical properties of a skew product with a curve of neutral points

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.

Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains

We consider a large class of piecewise expanding maps T of [0; 1] with a neutral xed point, and their associated Markov chain Yi whose transition kernel is the PerronFrobenius 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 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(Yi) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f T i may belong to the domain of normal attraction of a stable law of index p2 (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.

Optimal concentration inequalities for dynamical systems

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.

A Borel?Cantelli lemma for intermittent interval maps

We consider intermittent maps T of the interval, with an absolutely continuous invariant probability measure μ. Kim showed that there exists a sequence of intervals An such that ∑ μ(An) = ∞, but {An} does not satisfy the dynamical Borel–Cantelli lemma, i.e. for almost every x, the set {n : T (x) ∈ An} is finite. If ∑ Leb(An) = ∞, we prove that {An} satisfies the Borel–Cantelli lemma. Our results apply in particular to some maps T whose correlations are not summable. Mathematics Subject Classification: 37A25, 37C30, 37E05We consider intermittent maps T of the interval, with an absolutely continuous invariant probability measure μ. Kim showed that there exists a sequence of intervals An such that ∑ μ(An) = ∞, but {An} does not satisfy the dynamical Borel–Cantelli lemma, i.e. for almost every x, the set {n: Tn(x) An} is finite. If ∑ Leb(An) = ∞, we prove that {An} satisfies the Borel–Cantelli lemma. Our results apply in particular to some maps T whose correlations are not summable.

Local limit theorem for symmetric random walks in Gromov-hyperbolic groups

Completing a strategy of Gouezel 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

Limit theorems for coupled interval maps

R

Regularity of coboundaries for nonuniformly expanding Markov maps

the inverse of the spectral radius of the random walk, the probability to return to the identity at time