Recurrence times and rates of mixing
TL;DR: In this paper, the authors considered the setting of a map making "nice" return to a reference set, and defined criteria for the existence of equilibria, speed of convergence to equilibrium, and central limit theorem in terms of the tail of the return time function.
Abstract: 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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TL;DR: The aim of this work is to provide the readers with the know how for the application of recurrence plot based methods in their own field of research, and detail the analysis of data and indicate possible difficulties and pitfalls.
2,993 citations
Cites background from "Recurrence times and rates of mixin..."
...They have been linked to rates of mixing [40], and the relationship between the return time statistics of continuous and discrete systems has been investigated [41]....
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TL;DR: In this article, the concept of fractional kinetics is reviewed for systems with Hamiltonian chaos, where the notions of dynamical quasi-traps, Poincare recurrences, Levy flights, exit time distributions, phase space topology, etc.
1,478 citations
TL;DR: In this article, the authors review various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics, and adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution.
Abstract: This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures In terms of these one can prove the Gallavotti–Cohen fluctuation theorem One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations At equilibrium one recovers in particular the Onsager reciprocity relations Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss The results just mentioned hold so far only for classical systems; they do not involve large size, ie, they hold without a thermodynamic limit
436 citations
Cites background from "Recurrence times and rates of mixin..."
...Young [99]. The interest of the geometric approach is that it gives more detailed results than the general Pesin theory: the existence of an SRB measure is proved instead of being assumed and the (usually exponential) decay of correlations can be studied, see Young [97], [ 98 ]....
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TL;DR: 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.
Abstract: We present an original approach which allows us to investigate the statistical properties of a non-uniformly hyperbolic map on the interval. Based on a stochastic approximation of the deterministic map, 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.
435 citations
TL;DR: In this article, a generalized thermodynamic formalism for topological Markov shifts with a countable number of states was established and a definition of topological pressure was proposed. But the authors only consider the case where the equilibrium measure is a Gibbs measure.
Abstract: We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of . We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), where .
418 citations
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01 Jan 1987
TL;DR: In this article, the Brin-Katok Local Entropy Formula (LFP) was proposed for measure-preserving maps, which is a generalization of the Entropy of Expanding Maps.
Abstract: 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. The Poincare Recurrence Theorem.- 3. Volume-Preserving Diffeomorphisms and Flows.- 4. First Integrals.- 5. Hamiltonians.- 6. Continued Fractions.- 7. Topological Groups, Lie Groups, Haar Measure.- 8. Invariant Measures.- 9. Uniquely Ergodic Maps.- 10. Shifts: the Probabilistic Viewpoint.- 11. Shifts: the Topological Viewpoint.- 12. Equivalent Maps.- II. Ergodicity.- 1. Birkhoff's Theorem.- 2. Ergodicity.- 3. Ergodicity of Homomorphisms and Translations of the Torus.- 4. More Examples of Ergodic Maps.- 5. The Theorem of Kolmogorov-Arnold-Moser.- 6. Ergodic Decomposition of Invariant Measures.- 7. Furstenberg's Example.- 8. Mixing Automorphisms and Lebesgue Automorphisms.- 9. Spectral Theory.- 10. Gaussian Shifts.- 11. Kolmogorov Automorphisms.- 12. Mixing and Ergodic Markov Shifts.- III. Expanding Maps and Anosov Diffeomorphisms.- 1. Expanding Maps.- 2. Anosov Diffeomorphisms.- 3. Absolute Continuity of the Stable Foliation.- IV. Entropy.- 1. Introduction.- 2. Proof of the Shannon-McMillan-Breiman Theorem.- 3. Entropy.- 4. The Kolmogorov-Sinai Theorem.- 5. Entropy of Expanding Maps.- 6. The Parry Measure.- 7. Topological Entropy.- 8. The Variational Property of Entropy.- 9. Hyperbolic Homeomorphisms.- 10. Lyapunov Exponents. The Theorems of Oseledec and Pesin.- 11. Proof of Oseledec's Theorem.- 12. Proof of Ruelle's Inequality.- 13. Proof of Pesin's Formula.- 14. Entropy of Anosov Diffeomorphisms.- 15. Hyperbolic Measures. Katok's Theorem.- 16. The Brin-Katok Local Entropy Formula.- Notation Index.
941 citations
TL;DR: In this paper, a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains was proved under virtually no assumptions other than the necessary ones, and they used these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.
Abstract: 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 to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.
909 citations
TL;DR: In this article, the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces is discussed.
Abstract: 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 are for discrete time systems. To put this work into context, recall that for Axiom A attractors the picture has been fairly complete since the 1970’s (see [S1], [B], [R2]). Since then much progress has been made on two fronts: there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero Lyapunov exponents ([O], [P1], [Ka], [LY]), and there are detailed analyses of specific kinds of dynamical systems including, for example, billiards, 1-dimensional and Henon-type maps ([S2], [BSC]; [HK], [J]; [BC2], [BY1]). Statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero Lyapunov exponents. The goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than Axiom A. This class will not be defined explicitly, but it includes some of the much studied examples. By looking at regular returns to sets with good hyperbolic properties, one could give systems in this class a simple dynamical representation. Conditions for the existence of natural invariant measures, exponential mixing and central limit theorems are given in terms of the return times. These conditions can be checked in concrete situations, giving a unified way of proving a number of results, some new and some old. Among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of Henon-type maps.
875 citations
TL;DR: 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.
Abstract: We present an original approach which allows us to investigate the statistical properties of a non-uniformly hyperbolic map on the interval. Based on a stochastic approximation of the deterministic map, 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.
435 citations
TL;DR: In this article, Boyland, Boyland et al. presented a paper dedicated to Micheline Ishay, who would like to thank P. Collet and M. Wojtkowski for introducing me to the world of cones.
Abstract: *Dedicated to Micheline Ishay I would like to thank P. Boyland, L. Chierchia, V. Donnay, G. De Martino, C. Gole, J. L. Lebowitz, M. Lyubich, M. Rychlik, I. G. Schwarz, S. Vaienti and especially G. Gallavotti for helpful and enlightening discussions. Particularly warm thanks go to N. Chernov for carefully reading and finding a mistake in an early version; the present paper benefits from several improvements due to his sharp criticism. In addition, I am indebted to P. Collet and, most of all, M. Wojtkowski for introducing me to the magical world of cones. Finally, I thank J. Milnor, director of the Institute for Mathematical Sciences at Stony Brook University, where I was visiting during part of this work, and the Italian C.N.R.-GNFM for providing travel funds.
388 citations