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}
}
  • L. Young
  • Published 1 November 1999
  • Mathematics, Economics
  • Israel Journal of Mathematics
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. 
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References

SHOWING 1-10 OF 22 REFERENCES
Markov extensions and decay of correlations for certain Hénon maps
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
TLDR
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
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
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
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’
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
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
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
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.
  • P. Walters
  • Mathematics
    Ergodic 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
...
1
2
3
...