Results 1  10
of
12
RuellePerronFrobenius Spectrum For Anosov Maps
 Nonlinearity
, 2001
"... We extend a number of results from one dimensional dynamics based on spectral properties of the RuellePerronFrobenius transfer operator to Anosov di#eomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show ..."
Abstract

Cited by 33 (9 self)
 Add to MetaCart
We extend a number of results from one dimensional dynamics based on spectral properties of the RuellePerronFrobenius transfer operator to Anosov di#eomorphisms 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. 1.
Invariant Measures and Their Properties. A Functional Analytic Point of View
, 2002
"... In this series of lectures I try to illustrate systematically what I call the \functional analytic approach" to the study of the statistical properties of Dynamical Systems. The ideas are presented via a series of examples of increasing complexity, hoping to give in this way a feeling of the brea ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
In this series of lectures I try to illustrate systematically what I call the \functional analytic approach" to the study of the statistical properties of Dynamical Systems. The ideas are presented via a series of examples of increasing complexity, hoping to give in this way a feeling of the breadth of the method.
Orthonormal Expansions of Invariant Densities for Expanding Maps
 Adv. Math
, 2002
"... We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [U] of phase space discretisation. ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [U] of phase space discretisation.
QUASIINVARIANT MEASURES, ESCAPE RATES AND THE EFFECT OF THE HOLE
, 906
"... perturbation of T into an interval map with a hole. Given a number ℓ, 0 < ℓ < 1, we compute an upperbound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than −ln(1 − ℓ). The two main ingredients of our ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
perturbation of T into an interval map with a hole. Given a number ℓ, 0 < ℓ < 1, we compute an upperbound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than −ln(1 − ℓ). The two main ingredients of our approach are Ulam’s method and an abstract perturbation result of Keller and Liverani. 1.
Ulam's Scheme Revisited: Digital Modeling of Chaotic Attractors Via MicroPerturbations
, 2002
"... We consider discretizations fN of expanding maps f : I ! I in the strict sense: i.e. we assume that the only information available on the map is a nite set of integers. Using this de nition for computability, we show that by adding a random perturbation of order 1=N , the invariant measure corr ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We consider discretizations fN of expanding maps f : I ! I in the strict sense: i.e. we assume that the only information available on the map is a nite set of integers. Using this de nition for computability, we show that by adding a random perturbation of order 1=N , the invariant measure corresponding to f can be approximated and we can also give estimates of the error term. We prove that the randomized discrete scheme is equivalent to Ulam's scheme applied to the polygonal approximation of f , thus providing a new interpretation of Ulam's scheme. We also compare the eciency of the randomized iterative scheme to the direct solution of the N N linear system.
Ergodic Optimization
"... Let f be a realvalued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic faverage is as large as possible. In these notes we establish some basic aspects of the theory: equivalent definition ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Let f be a realvalued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic faverage is as large as possible. In these notes we establish some basic aspects of the theory: equivalent definitions of the maximum ergodic average, existence and generic uniqueness of maximizing measures, and the fact that every ergodic measure is the unique maximizing measure for some continuous function. Generic properties of the support of maximizing measures are described in the case where the dynamics is hyperbolic. A number of problems are formulated.
Computing The Rate Of Decay Of Correlations In Expanding And Hyperbolic Systems
, 2001
"... I discuss a general approach allowing to accurately investigative the statistical properties of expanding and hyperbolic dynamical systems. 1. Different approaches The study of the statistical properties of dynamical systems, and in particular the study of the rate of convergence to equilibrium has ..."
Abstract
 Add to MetaCart
I discuss a general approach allowing to accurately investigative the statistical properties of expanding and hyperbolic dynamical systems. 1. Different approaches The study of the statistical properties of dynamical systems, and in particular the study of the rate of convergence to equilibrium has a rather long story. In the last years several approaches have been developed, here is a brief (and incomplete) summary Coding the system: { Markov Partition (this is the original one, see [3] for an overview); { Towers (in the same spirit of coding the systems but much more exible [17]).
Phase Transition and Correlation Decay in Coupled Map Lattices
, 906
"... For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya’s probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoup ..."
Abstract
 Add to MetaCart
For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya’s probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoupling originally developed for weak coupling. This implies the exponential decay, in space and in time, of the correlation functions of the invariant measures. 1
Statistical properties of dynamical . . .
, 2011
"... We survey an area of recent development, relating dynamics to theoretical computer science. We discuss some aspects of the theoretical simulation and computation of the long term behavior of dynamical systems. We will focus on the statistical limiting behavior and invariant measures. We present a ge ..."
Abstract
 Add to MetaCart
We survey an area of recent development, relating dynamics to theoretical computer science. We discuss some aspects of the theoretical simulation and computation of the long term behavior of dynamical systems. We will focus on the statistical limiting behavior and invariant measures. We present a general method allowing the algorithmic approximation at any given accuracy of invariant measures. The method can be applied in many interesting cases, as we shall explain. On the other hand, we exhibit some examples where the algorithmic approximation of invariant measures is not possible. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem).
COMPUTATION OF SELBERG ZETA FUNCTIONS ON HECKE TRIANGLE GROUPS
, 804
"... ABSTRACT. In this paper, a heuristic method to compute the Selberg zeta function for Hecke triangle groups, Gq is described. The algorithm is based on the transfer operator method and an overview of the relevant background is given.We give numerical support for the claim that the method works and ca ..."
Abstract
 Add to MetaCart
ABSTRACT. In this paper, a heuristic method to compute the Selberg zeta function for Hecke triangle groups, Gq is described. The algorithm is based on the transfer operator method and an overview of the relevant background is given.We give numerical support for the claim that the method works and can be used to compute the Selberg Zeta function on Gq to any desired precision. We also present some numerical results obtained by implementing the algorithm. CONTENTS