Carlangelo Liverani

Decay of correlations

*Dedicated to Micheline Ishay I would like to thank P. Boyland, L. Chierchia, V. Donnay, G. De Martino, C. Gole, J. L. Lebowitz, M. Lyubich, M. Rychlik, I. G. Schwarz, S. Vaienti and especially G. Gallavotti for helpful and enlightening discussions. Particularly warm thanks go to N. Chernov for carefully reading and finding a mistake in an early version; the present paper benefits from several improvements due to his sharp criticism. In addition, I am indebted to P. Collet and, most of all, M. Wojtkowski for introducing me to the magical world of cones. Finally, I thank J. Milnor, director of the Institute for Mathematical Sciences at Stony Brook University, where I was visiting during part of this work, and the Italian C.N.R.-GNFM for providing travel funds.

A Probabilistic Approach To Intermittency

We present an original approach which allows to investigate the statistical properties of a non-uniform hyperbolic map of the interval. Based on a stochastic approximation of the deterministic map, this method gives essentially the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral xed point.

Ruelle–Perron–Frobenius spectrum for Anosov maps

We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle–Perron–Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows us to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasi-compact. (Information on the existence of a Sinai–Ruelle–Bowen measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated with smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows us to obtain easily very strong spectral stability results, which, in turn, imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam-type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite-dimensional problem.

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.).

Ergodicity in Hamiltonian Systems

We discuss the Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (nonuniform) hyperbolic behavior.

Decay of correlations for piecewise expanding maps

This paper investigates the decay of correlations in a large class of non-Markov one-dimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained.

Stability of statistical properties in two-dimensional piecewise hyperbolic maps

We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of two-dimensional maps with uniformly bounded second derivative, but we are confident that the present approach can be successful in much greater generality (we hope including higher dimensional billiards). For the class of systems at hand, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations, including deterministic and random perturbations and holes.

Rare Events, Escape Rates and Quasistationarity: Some Exact Formulae

We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging from the metric theory of continuos fractions and the Shannon capacity of constrained systems to the decay rate of metastable states, are given.

Random walk in Markovian environment

We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on

Rigorous numerical investigation of the statistical properties of piecewise expanding maps. A feasibility study

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