SRB measures for partially hyperbolic systems whose central direction is mostly expanding

Alves, José F.; Bonatti, Christian; Viana, Marcelo

doi:10.1007/978-0-387-21830-4_24

Cited by 86 publications

(104 citation statements)

References 23 publications

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“…Indeed, the assumptions imply that the graph transform of F k converges to the graph transform of F in an appropriate sense, so that the corresponding fixed points converge as well. This yields (1) and (2). Part (3) is a direct consequence of (2) and the definition (35), in the case y ∈ W s ε (y).…”

Section: Continuity Of Holonomiesmentioning

confidence: 71%

“…Let P :Ê →M be a measurable fiber bundle with fibers modeled on N . By this we meanÊ comes with a countable system of bijections (1)P −1 (U n ) → U n × N that map each fiberÊx =P −1 (x) onto {x} × N , and all coordinate changes are measurable maps of the form (2) (…”

Section: Smooth Cocyclesmentioning

confidence: 99%

“…When the vectors in the center bundle E c also have non-zero Lyapunov exponents, that is, when lim 1 n log Df n (x)v = 0 ∀v ∈ E c x at typical points x ∈ M , one can build on (non-uniform) hyperbolicity theory to derive important geometric and statistical information on the dynamics. See Pesin [38,39], Ledrappier, Young [32,34,35], Katok [29], Barreira, Pesin, Schmeling [8], Young [45,46], and Alves, Bonatti, Viana [2,15].…”

Section: Introductionmentioning

confidence: 99%

“…The Invariance Principle allows them to treat the case of vanishing center Lyapunov exponent, which was not covered by [2,15]. Also very recently, F. and J. RodriguezHertz, R. Ures, A. Tahzibi announced further applications of these methods, to entropy maximizing measures of partially hyperbolic maps.…”

Section: Introductionmentioning

confidence: 99%

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Extremal Lyapunov exponents: an invariance principle and applications

2010

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Abstract. We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic ("center") behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate its power.We find that any ergodic automorphism of the 4-torus with two eigenvalues in the unit circle is stably Bernoulli among symplectic maps. Indeed, any nearby symplectic map has no zero Lyapunov exponents, unless it is volume preserving conjugate to the automorphism itself. Another main application is to accessible skew-product maps preserving area on the fibers. We prove, in particular, that if the genus of the fiber is at least 2 then the Lyapunov exponents must be different from zero and vary continuously with the map.These, and other dynamical conclusions, originate from a general Invariance Principle we prove in here. It is formulated in terms of smooth cocycles, that is, fiber bundle morphisms acting by diffeomorphisms on the fibers. The extremal Lyapunov exponents measure the smallest and largest exponential rates of growth of the derivative along the fibers. The Invariance Principle states that if these two numbers coincide then the fibers carry some amount of structure which is transversely invariant, that is, invariant under certain canonical families of homeomorphisms between fibers.

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Section: Continuity Of Holonomiesmentioning

confidence: 71%

Section: Smooth Cocyclesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Extremal Lyapunov exponents: an invariance principle and applications

2010

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“…If the central direction is mostly contracting, Bonatti-Viana [22] showed that there are finitely many SRB measures, and Castro [31] and Dolgopyat [44] independently showed, under mild additional technical assumptions, that correlation functions (for Hölder observables) decay exponentially fast. When the central direction is mostly expanding, Alves-Bonatti-Viana [3] proved that there exist SRB measures (and finitely many such measures if the Lyapunov exponents are bounded away from zero).…”

Section: Partially Hyperbolic Diffeomorphismsmentioning

confidence: 99%

Spectrum and Statistical Properties of Chaotic Dynamics

Baladi

2001

European Congress of Mathematics

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Abstract. We present new developments on the statistical properties of chaotic dynamical systems. We concentrate on the existence of an ergodic physical (SRB) invariant measure and its mixing properties, in particular decay of its correlation functions for smooth observables. In many cases, there is a connection (via the spectrum of a Ruelle-Perron-Frobenius transfer operator) with the analytic properties of a weighted dynamical zeta function, weighted dynamical Lefschetz function, or dynamical Ruelle-Fredholm determinant, built using the periodic orbit structure of the map.

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Physical measures for partially hyperbolic surface endomorphisms

Tsujii

2005

Acta Math.

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SRB measures for partially hyperbolic systems whose central direction is mostly expanding

Cited by 86 publications

References 23 publications

Extremal Lyapunov exponents: an invariance principle and applications

Extremal Lyapunov exponents: an invariance principle and applications

Spectrum and Statistical Properties of Chaotic Dynamics

Physical measures for partially hyperbolic surface endomorphisms

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