Recurrence times and rates of mixing
@article{Young1999RecurrenceTA, title={Recurrence times and rates of mixing}, author={Lai-Sang Young}, journal={Israel Journal of Mathematics}, year={1999}, volume={110}, pages={153-188} }
The setting of this paper consists of a map making “nice” returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem are given in terms of the tail of the return time function. The abstract setting considered arises naturally in differentiable dynamical systems with some expanding or hyperbolic properties.
612 Citations
Recurrence times and rates of mixing for invertible dynamical systems
- Mathematics
- 2006
We consider invertible discrete-time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable recurrence time. We show that…
RECURRENCE RATE IN RAPIDLY MIXING DYNAMICAL SYSTEMS
- Mathematics
- 2004
For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point.
We prove that when the decay of correlation is…
Slow Rates of Mixing for Dynamical Systems with Hyperbolic Structures
- Mathematics
- 2008
We consider invertible discrete-time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable return time. We show that the…
On mixing and the local central limit theorem for hyperbolic flows
- MathematicsErgodic Theory and Dynamical Systems
- 2020
We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal…
Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains
- Mathematics
- 2003
We present a spectral theory for a class of operators satisfying a weak “Doeblin–Fortet" condition and apply it to a class of transition operators. This gives the convergence of the series ∑k≥0 kr Pk…
Mixing for continuous time dynamical systems with infinite measure
- Mathematics
- 2013
We develop operator renewal theory for flows and apply this to infinite ergodic theory. In particular we obtain results on mixing for a large class of infinite measure semiflows.
Examples of systems…
A note on mixing in interval maps
- Mathematics, Physics
- 2013
We construct a class of simple dynamical systems for which all correlation properties, i.e., the entire spectrum of the Perron-Frobenius operator, is accessible by analytical means. As an application…
Rates in almost sure invariance principle for quickly mixing dynamical systems
- MathematicsStochastics and Dynamics
- 2019
For a large class of quickly mixing dynamical systems, we prove that the error in the almost sure approximation with a Brownian motion is of order [Formula: see text] with [Formula: see text].…
Convergence rate of the powers of an operator. Applications to stochastic systems
- Mathematics
- 2017
We extend the traditional operator theoretic approach for the study of dynamical systems in order to handle the problem of non-geometric convergence. We show that the probabilistic treatment…
Lyapunov exponents and rates of mixing for one-dimensional maps
- Mathematics, PhysicsErgodic Theory and Dynamical Systems
- 2004
We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is…
References
SHOWING 1-10 OF 22 REFERENCES
Markov extensions and decay of correlations for certain Hénon maps
- Mathematics
- 2000
Henon maps for which the analysis in [BC2] applies are considered. Sets with good hyperbolic properties and nice return structures are constructed and their return time functions are shown to have ...
A probabilistic approach to intermittency
- Mathematics, Computer ScienceErgodic Theory and Dynamical Systems
- 1999
This method essentially gives the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral fixed point.
Ergodic Properties of Invariant Measures for Piecewise Monotonic Transformations
- Mathematics
- 1982
During the last decade a lot of research has been done on onedimensional dynamics. Different kinds of invariant measures for certain classes of piecewise monotonic transformations have been…
First return map and invariant measures
- Mathematics
- 1980
We give sufficient conditions for the existence of absolutely continuous invariant measures, finite or σ-finite, for maps on the interval. We givea priori bound for the number of different ergodic…
Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions
- Mathematics
- 1986
We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results…
Nonexistence of SBR measures for some diffeomorphisms that are ‘Almost Anosov’
- MathematicsErgodic Theory and Dynamical Systems
- 1995
Abstract The purpose of this paper is to present some simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Each example has a fixed point at which…
Rates of mixing for potentials of summable variation
- Mathematics
- 1999
It is well known that for subshifts of finite type and equilibrium measures associated to Hölder potentials we have exponential decay of correlations. In this article we derive explicit rates of…
STATISTICAL PROPERTIES OF DYNAMICAL SYSTEMS WITH SOME HYPERBOLICITY
- Mathematics
- 1998
This paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces. The main results…
Ergodic Theory and Differentiable Dynamics
- Mathematics
- 1986
0. Measure Theory.- 1. Measures.- 2. Measurable Maps.- 3. Integrable Functions.- 4. Differentiation and Integration.- 5. Partitions and Derivatives.- I. Measure-Preserving Maps.- 1. Introduction.- 2.…
Ergodic Theory and Differentiable Dynamics By Ricardo Mañé: Translated from the Portuguese by Silvio Levy. Ergebnisse de Mathematik und ihrer Grenzgebiete, 3 Folge-Band 8. Springer-Verlag 1987.
- MathematicsErgodic Theory and Dynamical Systems
- 1989
This book is the eagerly awaited translation into English of the I.M.P.A. monograph written in Portuguese and not easily available outside Brazil. The aim of the book is to describe the rudiments of…