zbMATH — the first resource for mathematics

Recurrence times and rates of mixing. (English) Zbl 0983.37005
Summary: 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.

37A25 Ergodicity, mixing, rates of mixing
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
Full Text: DOI
[1] [BY] M. Benedicks and L.-S. Young,Decay or correlations for certain Henon maps, to appear in Astérisque, in a volume in honor of Douady.
[2] [FL] A. Fisher and A. Lopes,Polynomial decay of correlation and the central limit theorem for the equilibrium state of a non-Hölder potential, preprint, 1997.
[3] [HK] F. Hofbauer and G. Keller,Ergodic properties of invariant measures for piecewise monotonic transformations, Mathematische Zeitschrift180 (1982), 119–140. · Zbl 0485.28016
[4] [H] H. Hu,Decay of correlations for piecewise smooth maps with indifferent fixed points, preprint. · Zbl 1071.37026
[5] [HY] H. Hu and L.-S. Young,Nonexistence of SBRmeasures for some systems that are ”almost Anosov”, Ergodic Theory and Dynamical Systems15 (1995), 67–76. · Zbl 0818.58035
[6] [I] S. Isola,On the rate of convergence to equilibrium for countable ergodic Markov chains, preprint, 1997.
[7] [KV] C. Kipnis and S.R.S. Varadhan,Central limit theorem for additive functions of reversible Markov process and applications to simple exclusions, Communications in Mathematical Physics104 (1986), 1–19. · Zbl 0588.60058
[8] [L1] C. Liverani,Decay of correlations, Annals of Mathematics142 (1995), 239–301. · Zbl 0871.58059
[9] [L2] C. Liverani,Central limit theorem for deterministic systems, International Conference on Dynamical Systems, Montevideo 1995 (F. Ledrappier, J. Lewowicz and S. Newhouse, eds.), Pitman Research Notes in Mathematics362 (1996), 56–75.
[10] [LSV] C. Liverani, B. Saussol and S. Vaienti,A probabilistic approach to intermittency, preprint.
[11] [M] R. Mañé,Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1983.
[12] [Pi] G. Pianigiani,First return maps and invariant measures, Israel Journal of Mathematics35 (1980), 32–48. · Zbl 0445.28016
[13] [Po] M. Pollicott,Rates of mixing for potentials of summable variation, to appear in Transactions of the American Mathematical Society.
[14] [Pt] J.W. Pitman,Uniform rates of convergence for Markov chain transition probabilities, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete29 (1974), 193–227. · Zbl 0373.60077
[15] [R] D. Ruelle,Thermodynamic Formalism, Addison-Wesley, New York, 1978.
[16] [TT] P. Tuominen and R. Tweedie,Subgeometric rates of convergence of f-ergodic Markov chains, Advances in Applied Probability26 (1994), 775–798. · Zbl 0803.60061
[17] [Y] L.-S. Young,Statistical properties of dynamical systems with some hyperbolicity, Annals of Mathematics147 (1998), 558–650. · Zbl 0945.37009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.