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93
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 ..."
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Cited by 260 (14 self)
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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 two-pronged: 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 Hénon-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. 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 ..."
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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...
CENTRAL LIMIT THEOREM FOR DETERMINISTIC SYSTEMS
, 1996
"... A unified approach to obtaining the central limit theorem for hyperbolic dynamical systems is presented. It builds on previous results for one dimensional maps but it applies to the multidimensional case as well. ..."
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Cited by 77 (4 self)
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A unified approach to obtaining the central limit theorem for hyperbolic dynamical systems is presented. It builds on previous results for one dimensional maps but it applies to the multidimensional case as well.
Ruelle-Perron-Frobenius Spectrum For Anosov Maps
- Nonlinearity
, 2001
"... We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov di#eomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show ..."
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Cited by 76 (17 self)
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We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov di#eomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem. 1.
On contact Anosov flows
- Ann. of Math
"... Abstract. Exponential decay of correlations for C 4 Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature. 1. ..."
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Cited by 72 (11 self)
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Abstract. Exponential decay of correlations for C 4 Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature. 1.
Absolutely Continuous Invariant Measures for Multidimensional Expanding Maps
, 1998
"... We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps w ..."
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Cited by 55 (2 self)
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We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps with singularities. We allow maps that are discontinuous on some extremely wild sets, the shape of the discontinuities being completely ignored with our approach. Key-Words: Expanding maps with singularities, Invariant measure, Decay of correlations, Perron-Frobenius operator. October 1997 Second version December 1998 CPT-97/P.3552 anonymous ftp : ftp.cpt.univ-mrs.fr web : www.cpt.univ-mrs.fr Unite Propre de Recherche 7061 1 PhyMat, Mathematics Department, University of Toulon. 83957 La Garde, France. Invariant Measure for Multidimensional Expanding Maps 1 Contents 1 Introduction 1 2 Piecewise expanding maps 4 3 Quasi-Holder space 9 4 A Lasota-Yorke type Inequality 12 5 Spectral res...
Lasota-Yorke Maps With Holes: Conditionally Invariant Probability Measures And Invariant Probability Measures On The Survivor Set
, 2001
"... Let T : I ! I be a Lasota-Yorke map on the interval I, let Y be a non trivial sub-interval of I and g , be a strictly positive potential which belongs to BV and admits a conformal measure m. We give constructive conditions on Y ensuring the existence of absolutely continuous (w.r.t. m) conditio ..."
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Cited by 35 (3 self)
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Let T : I ! I be a Lasota-Yorke map on the interval I, let Y be a non trivial sub-interval of I and g , be a strictly positive potential which belongs to BV and admits a conformal measure m. We give constructive conditions on Y ensuring the existence of absolutely continuous (w.r.t. m) conditionally invariant probability measures to non absorption in Y . These conditions imply also existence of an invariant probability measure on the set X1 of points which never fall into Y . Our conditions allow rather "large" holes.
Stability of statistical properties in two-dimensional piecewise hyperbolic maps Trans.
- Am. Math. Soc.
, 2008
"... Abstract. We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of two-dimensional maps with uniformly bounded second derivative. For ..."
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Cited by 34 (17 self)
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Abstract. We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of two-dimensional maps with uniformly bounded second derivative. For the class of systems at hand, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations, including deterministic and random perturbations and holes.
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.
