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48
Almost sure invariance principle for nonuniformly hyperbolic systems.
 Comm. Math. Phys.
, 2005
"... Abstract 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 fin ..."
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Cited by 63 (14 self)
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Abstract 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 limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.
Equilibrium states for interval maps: the potential −tlog Df
"... Abstract. We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials ϕ with the ‘bounded range ’ condition supϕ − inf ϕ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use ..."
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Cited by 36 (9 self)
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Abstract. We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials ϕ with the ‘bounded range ’ condition supϕ − inf ϕ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of PerronFrobenius operators. We demonstrate that this ‘bounded range ’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context. 1.
Nonuniform hyperbolicity in complex dynamics I,II
"... We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer nc so that ∑∞ n=1 (F n)′(F nc (c))  −α < ∞ and F has no parabolic periodic cycles. Let µmax be the maximal multipl ..."
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Cited by 32 (4 self)
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We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer nc so that ∑∞ n=1 (F n)′(F nc (c))  −α < ∞ and F has no parabolic periodic cycles. Let µmax be the maximal multiplicity of the critical points. The objective is to study the Poincaré series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. If F is summable with exponent α < exists a unique, ergodic, and nonatomic conformal measure ν with exponent δPoin(J) = HDim(J). If F is polynomially summable with the exponent α, ∑∞ n=1 n(F n)′(F nc (c))  −α < ∞ and F has no parabolic periodic cycles, then F has an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval.
Decay of Correlations for Lorentz Gases and Hard Balls
, 2000
"... We discuss rigorous results and open problems on the decay of correlations for dynamical systems characterized by elastic collisions: hard balls, Lorentz gases, Sinai billiards and related models. Recently developed techniques for general dynamical systems with some hyperbolic behavior are discuss ..."
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Cited by 31 (2 self)
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We discuss rigorous results and open problems on the decay of correlations for dynamical systems characterized by elastic collisions: hard balls, Lorentz gases, Sinai billiards and related models. Recently developed techniques for general dynamical systems with some hyperbolic behavior are discussed. These techniques give exponential decay of correlations for many classes of billiards and open the door to further investigations.
The Lorenz attractor is mixing
 Comm. Math. Phys
"... Abstract. We study a class of geometric Lorenz flows, introduced independently by Afraĭmovič, Bykov & Sil ′ nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing. ..."
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Cited by 27 (3 self)
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Abstract. We study a class of geometric Lorenz flows, introduced independently by Afraĭmovič, Bykov & Sil ′ nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.
EQUILIBRIUM MEASURES FOR MAPS WITH INDUCING SCHEMES
, 2008
"... Abstract. We introduce a class of continuous maps f of a compact topological space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, i.e., describe a class of realvalued potential functions ϕ on I, which admit a uniq ..."
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Cited by 24 (5 self)
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Abstract. We introduce a class of continuous maps f of a compact topological space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, i.e., describe a class of realvalued potential functions ϕ on I, which admit a unique equilibrium measure µϕ minimizing the free energy for a certain class of invariant measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the central limit theorem. Our results apply to certain maps of the interval with critical points and/or singularities (including some unimodal and multimodal maps) and to potential functions ϕt = −t log df  with t ∈ (t0, t1) for some t0 < 1 < t1. In the particular case of Sunimodal maps we show that one can choose t0 < 0 and that the class of measures under consideration consists of all invariant Borel probability measures. 1.
Statistical properties of topological ColletEckmann maps
 ANN. SCI. ÉCOLE NORM. SUP
, 2006
"... We study geometric and statistical properties of complex rational maps satisfying the Topological ColletEckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and that this measure is nonatomic, ergodic and that its Hausdorf ..."
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Cited by 21 (9 self)
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We study geometric and statistical properties of complex rational maps satisfying the Topological ColletEckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and that this measure is nonatomic, ergodic and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and we show that this measure is exponentially mixing (it has exponential decay of correlations) and that it satisfies the Central Limit Theorem. We also show that for a complex rational map f the existence of such an invariant measure characterizes the Topological ColletEckmann Condition, and that this measure is the unique equilibrium state with potential −HD(J(f)) ln f ′ .
Existence and convergence properties of physical measures for certain dynamical systems with holes
 ERGODIC THEORY DYNAM. SYS
, 2010
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Statistical properties of onedimensional maps with critical points and singularities
 Stoch. Dyn
"... Abstract. We prove that a class of onedimensional maps with an arbitrary number of nondegenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential d ..."
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Cited by 18 (5 self)
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Abstract. We prove that a class of onedimensional maps with an arbitrary number of nondegenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for Hölder observations.
Almost Sure Rates of Mixing for I.i.d. Unimodal Maps
, 1999
"... . It has been known since the pioneering work of Jakobson and subsequent work by BenedicksCarleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and KellerNowicki proved exponential decay of its correlation functions. BenedicksYou ..."
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Cited by 16 (3 self)
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. It has been known since the pioneering work of Jakobson and subsequent work by BenedicksCarleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and KellerNowicki proved exponential decay of its correlation functions. BenedicksYoung [BeY] and BaladiViana [BV] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical properties of the sample measures of i.i.d. itineraries are more dicult to estimate than the \averaged statistics." Adapting to random systems, on the one hand the notion of hyperbolic times due to Alves [A], and on the other a probabilistic coupling method introduced by Young [Yo2] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing. 1. Introduction An important class of discretetime deterministic dynamical systems (given by a transformation f on a Rieman...