Results 1  10
of
77
STATISTICAL PROPERTIES OF DYNAMICAL SYSTEMS WITH SOME HYPERBOLICITY
, 1997
"... 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 ..."
Abstract

Cited by 260 (14 self)
 Add to MetaCart
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 the development has, by and large, been twopronged: 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, 1dimensional and Hénontype 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. In this paper I will attempt to understand 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
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 ..."
Abstract

Cited by 238 (10 self)
 Add to MetaCart
(Show Context)
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...
Limit theorems for partially hyperbolic systems
 Trans. Amer. Math. Soc
"... Abstract. We consider a large class of partially hyperbolic systems containing, among others, ane maps, frame
ows on negatively curved manifolds and mostly contracting dieomorphisms. If the rate of mixing is suciently high the system satises many classical limit theorems of probability theory. 1. ..."
Abstract

Cited by 82 (14 self)
 Add to MetaCart
(Show Context)
Abstract. We consider a large class of partially hyperbolic systems containing, among others, ane maps, frame
ows on negatively curved manifolds and mostly contracting dieomorphisms. If the rate of mixing is suciently high the system satises many classical limit theorems of probability theory. 1. Introduction. The study of the statistical properties of deterministic systems constitutes an important branch of smooth ergodic theory. According to a modern view, a chaotic behavior of deterministic systems is caused by the exponential instability of nearby trajectories. The best illustra
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 ..."
Abstract

Cited by 63 (14 self)
 Add to MetaCart
(Show Context)
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.
Recent advances in invariance principles for stationary sequences.
 Probab. Surv.
, 2006
"... Abstract: In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The ..."
Abstract

Cited by 50 (7 self)
 Add to MetaCart
(Show Context)
Abstract: In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingalelike sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains. AMS 2000 subject classifications: 60G51, 60F05.
On mixing properties of compact group extensions of hyperbolic systems
 Israel J. Math
"... Abstract. We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is alw ..."
Abstract

Cited by 46 (7 self)
 Add to MetaCart
(Show Context)
Abstract. We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is always the case for ergodic semisimple extensions as well as for stably ergodic extensions of Anosov diffeomorphisms of infranilmanifolds. 1.
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 ..."
Abstract

Cited by 40 (13 self)
 Add to MetaCart
(Show Context)
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.
Statistical Limit Theorems for Suspension Flows
, 2004
"... In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function. ..."
Abstract

Cited by 39 (17 self)
 Add to MetaCart
In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function.
Decay of Correlations of the RauzyVeechZorich Induction Map on the Space of Interval Exchange Transformations
, 2005
"... The aim of this paper is to prove a stretchedexponential bound for the decay of correlations for the RauzyVeechZorich induction map on the space of interval exchange transformations (Theorem 4). A Corollary is the Central Limit Theorem for the Teichmüller flow (Theorem 10). The proof ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
(Show Context)
The aim of this paper is to prove a stretchedexponential bound for the decay of correlations for the RauzyVeechZorich induction map on the space of interval exchange transformations (Theorem 4). A Corollary is the Central Limit Theorem for the Teichmüller flow (Theorem 10). The proof