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344
Recurrence Times And Rates Of Mixing
, 1997
"... 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 a ..."
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Cited by 238 (10 self)
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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. This paper is part of an attempt to understand the speed of mixing and related statistical properties for chaotic dynamical systems. More precisely, we are interested in systems that are expanding or hyperbolic on large parts (though not necessarily all) of their phase spaces. A natural approach to this problem is to pick a suitable reference set, and to regard a part of the system as having "renewed" itself when it makes a "full" return to this set. We obtain in this way a representation of the dynamical system in question, described in terms of a reference set and return times. We propose to study thi...
What are SRB measures, and which dynamical systems have them?
"... This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle and Bowen in the 1970s. SRB measures ..."
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Cited by 130 (12 self)
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This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle and Bowen in the 1970s. SRB measures, as these objects are called, play an important role in the ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly speaking, • SRB measures are the invariant measures most compatible with volume when volume is not preserved; • they provide a mechanism for explaining how local instability on attractors can produce coherent statistics for orbits starting from large sets in the basin. An outline of this paper is as follows. The original work of Sinai, Ruelle and Bowen was carried out in the context of Anosov and Axiom A systems. For these dynamical systems they identified and constructed an invariant measure which is uniquely important from several different points of view. These pioneering works are reviewed in Section 1. Subsequently, a nonuniform, almosteverywhere notion of hyperbolicity expressed in terms of Lyapunov exponents was developed. This notion provided a new framework for the ideas in the last paragraph. While not all of the previous characterizations are equivalent in this broader setting, the central ideas have remained intact, leading to a more general notion of SRB measures. This is discussed in Section 2. The extension above opened the door to the possibility that the dynamics on many attractors are described by SRB measures. Determining if this is (or is not) the case, however, let alone proving it, has turned out to be very challenging. No genuinely nonuniformly hyperbolic examples were known until the early 1990s, when SRB measures were constructed for certain Hénon maps. Today we still do not have a good understanding of which dynamical systems admit SRB measures, but some progress has been made; a sample of it is reported in Section 3.
Dimensions and measures in infinite iterated function systems
 PROC. LONDON MATH. SOC
, 1996
"... The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to ..."
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Cited by 114 (25 self)
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The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to be unique. The existence of a unique invariant probability equivalent to the conformal measure is derived. Our methods employ the concepts of the PerronFrobenius operator, symbolic dynamics on an infinite dimensional shift space, and the properties of the above mentioned ergodic invariant measure. A formula for the Hausdorff dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by finite systems are shown to appear in the infinite case. In particular a variety of conditions are provided for Hausdorff and packing measures to be positive or finite, and a number of examples are described showing the appearance of various possible combinations for these quantities. One example given special attention is the limit set associated to the complex continued fraction expansion  in particular lower and upper estimates for its Hausdor dimension are given. A large natural class of systems whose limit sets are "dimensionless in the restricted sense" is described.
Markov extensions and decay of correlations for certain Hénon maps, Astérisque
, 2000
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Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems
, 2005
"... We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite hori ..."
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Cited by 66 (14 self)
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We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon. Statistical
Dynamical Sources in Information Theory: A General Analysis of Trie Structures
 ALGORITHMICA
, 1999
"... Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of arraytries, list tries, and bsttries ("ternary search tries"). The size an ..."
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Cited by 60 (7 self)
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Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of arraytries, list tries, and bsttries ("ternary search tries"). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of height of tries is given. The unifying data model used is that of dynamical sources and it encompasses classical models like those of memoryless sources with independent symbols, of finite Markovchains, and of nonuniform densities. The probabilistic behaviour of the main parameters, namely size, path length, or height, appears to be determined by two intrinsic characteristics of the source: the entropy and the probability of letter coincidence. These characteristics are themselves related in a natural way to spectral properties of specific transfer operators of the Ruelle type.
High Temperature Expansions and Dynamical Systems
 Comm. Math. Phys
, 1996
"... We develop a resummed hightemperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the Pe ..."
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Cited by 48 (3 self)
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We develop a resummed hightemperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the PerronFrobenius operator of weakly coupled map lattices. 1 Introduction. The theory of Gibbs states was originally developed for the mathematical analysis of equilibrium statistical mechanics. An interesting application of the theory was found by Sinai, Ruelle and Bowen in the 70's [41, 42, 38, 1] who applied it to the ergodic theory of uniformly hyperbolic dynamical systems. While this so called thermodynamic formalism has been very successful in ergodic theory, the Gibbs states that describe the statistics of such dynamical systems are quite simple from the point of view of statistical mechanics: they describe one dimensional spin systems with spins taking values in a finite set and intera...
A multifractal analysis of equilibrium measures for conformal expanding maps and Markov Moran geometric constructions
, 1997
"... Abstract. In this paper we establish the complete multifractal formalism for equilibrium measures for Holder continuous conformal expanding maps and expanding Markov Moranlike geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps wi ..."
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Cited by 43 (2 self)
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Abstract. In this paper we establish the complete multifractal formalism for equilibrium measures for Holder continuous conformal expanding maps and expanding Markov Moranlike geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Holder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails. Invariant sets of most dynamical systems in general are not selfsimilar in the strict sense. However, part of these sets can sometimes be decomposed into (perhaps uncountably many) subsets each supporting a Borel probability measure possessing a type of scaling symmetry. This means that the measure admits a group of scale symmetries which reproduces copies
Decay of correlations, Central limit theorems and approximation by brownian motion for compact Lie group extensions
 Ergod. Th. & Dynam. Sys
, 2003
"... Hölder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion). Using an equivariant version of the Ruelle transfer operator studied by Parry ..."
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Cited by 40 (12 self)
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Hölder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion). Using an equivariant version of the Ruelle transfer operator studied by Parry & Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.