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Carlangelo Liverani

Publications and source records attributed to Carlangelo Liverani.

48 records · Page 3Linked to original sources
arXiv2008

Random walk in Markovian environment

We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on $\mathbb{Z}^d$. We assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is Markovian with strong spatial and temporal mixing properties.

math.PR↗
arXiv2008

Random Walk in deterministically changing environment

We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT.

math.PR↗
arXiv2007

Quenched CLT for random toral automorphism

We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps.

math.DS↗
arXiv2007

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.

math.DS↗
arXiv2006

Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties

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.

math.DS↗
arXiv2006

Zeta functions and Dynamical Systems

In this brief note we present a very simple strategy to investigate dynamical determinants for uniformly hyperbolic systems. The construction builds on the recent introduction of suitable functional spaces which allow to transform simple heuristic arguments in rigorous ones. Although the results so obtained are not exactly optimal the straightforwardness of the argument makes it noticeable.

math.DS↗
arXiv2005

Fredholm determinants, Anosov maps and Ruelle resonances

I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for $\Co^\infty$ Anosov diffeomorphisms, the dynamical determinant is an entire function.

math.DS↗
arXiv2005

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^\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.).

math.DS↗
arXiv2001

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 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 quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows 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.

nlin.CD↗