Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities (original) (raw)

Large deviations in non-uniformly hyperbolic dynamical systems

Ergodic Theory and Dynamical Systems, 2008

We prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

The Bernoulli Property for Weakly Hyperbolic Systems

Journal of Statistical Physics, 2004

A dynamical system is called partially hyperbolic if it exhibits three invariant directions, one unstable (expanding), one stable (contracting) and one central direction (somewhere in between the other two). We prove that topologically mixing partially hyperbolic diffeomorphisms whose central direction is non-uniformly contracting (negative Lyapunov exponents) almost everywhere have the Bernoulli property: the system is equivalent to an i. i. d. (independently identically distributed) random process. In particular, these systems are mixing: correlations of integrable functions go to zero as time goes to infinity. We also extend this result in two different ways. Firstly, for 3-dimensional diffeomorphisms, if one requires only non-zero (instead of negative) Lyapunov exponents then one still gets a quasi-Bernoulli property. Secondly, if one assumes accessibility (any two points are joined by some path whose legs are stable segments and unstable segments) then it suffices to requires the mostly contracting property on a positive measure subset, to obtain the same conclusions.

Fine ergodic properties of partially hyperbolic dynamical systems

Let f : T 3 ! T 3 be a C 2 volume preserving partially hyperbolic diffeomorphism homotopic to a linear Anosov automorphism A : T 3 ! T 3. We prove that if f is Kolmogorov, then f is Bernoulli. We study the characteristics of atomic disintegration of the volume measure whenever it occurs. We prove that if the volume measure m has atomic disintegration on the center leaves then the disintegration has one atom per center leaf. We give a condition, depending only on the center Lyapunov exponent of the diffeomorphism, that guarantees atomic disintegration of the volume measure on center leaves. We construct an open family of diffeomorphisms satisfying this condition which generates the first examples of foliations which are both measurable and minimal. In this same construction we give the first examples of partially hyperbolic diffeomorphisms with zero center Lyapunov exponent and homotopic to a linear

A Local Ergodic Theorem for Non-Uniformly Hyperbolic Symplectic Maps with Singularities

2016

Abstract. In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski. In this paper, we consider a class of invertible maps with discontinuities and unbounded derivatives, which model billiards and other physical systems. Let F be one of these maps, and assume that F preserves a symplectic form and is non-uniformly hyperbolic. The last condition means that the Lyapunov exponents of F are non-zero with respect to the symplectic volume, which is invariant. Our main result (Theorem 3.1) establishes sufficient conditions for a point in the domain of F to have a neighborhood contained up to a set of zero measure in one ergodic component of F. In fact, we prove a stronger result, namely, that the mentioned neighborhood is contained up to a set of zero measure in one Bernoulli component of F. Results of this type are often called ‘local ergodic theorems’. Loc...

On the ergodicity of partially hyperbolic systems

Annals of Mathematics, 2010

Pugh and Shub have conjectured that essential accessibility implies ergodicity for a C 2 , partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for C 1Cı partially hyperbolic systems.

Notes on Ergodicity of Partially Hyperbolic Systems

2008

These lecture notes were written to accompany a minicourse given at the School and Workshop on Dynamical Systems at IMPA (Rio de Janeiro), January 2006. Warning: These notes are unedited lecture notes and do contain errors! Proceed with caution... Let f : M → M be a (pointwise) partially hyperbolic diffeomor-phism. Recall what partially hyperbolic means. At every point in M , there exist tangent vectors that are uniformly contracted by the derivative T f and tangent vectors that are uniformly expanded by T f. The contracted vectors lie in an invariant subbundle E s of T M , called the stable subbundle, and the expanded vectors lie in an invariant subbun-dle E u , called the unstable subbundle. The sum of these subbundles E u ⊕ E s is the hyperbolic part of T M. The rest of T M is also under control: there is a T f-invariant complement E c to E u ⊕ E s in T M , called the center bundle. Tangent vectors in E c may be expanded or contracted under the action of T f , but they are neithe...

Stably Ergodic Dynamical Systems and Partial Hyperbolicity

Journal of Complexity, 1997

In this paper we show that a little hyperbolicity goes a long way toward guaranteeing stable ergodicity, and in fact may be necessary for it. Our main theorem may be interpreted as saying that the same phenomenon producing chaotic behavior (i.e., some hyperbolicity) also leads to robust statistical behavior. Examples to which our theory applies include translations on certain homogeneous spaces and the time-one map of the geodesic flow for a manifold of constant negative curvature.