Journal ArticleDOI
The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin's entropy formula
TLDR
In this article, a formule valable for toutes les mesures invariant to Margulis et Ruelle's inequality is defined, and demontre une formule with respect to all the invariant mesures.Abstract:
Soit M une variete de Riemann compacte, soit f:M→M un diffeomorphisme, et soit m une mesure de probabilite de Borel f-invariante sur M. On identifie les mesures pour lesquelles l'inegalite de Margulis et Ruelle qui relie l'entropie aux exposants de Lyapunov atteint l'egalite. On demontre une formule valable pour toutes les mesures invariantesread more
Citations
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Journal ArticleDOI
What Are SRB Measures, and Which Dynamical Systems Have Them?
TL;DR: In this article, the authors reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle, and Bowen in the 1970s, called SRB measures, as these objects are called.
Journal ArticleDOI
Invariant measures and arithmetic quantum unique ergodicity
TL;DR: In this paper, the authors classify measures on the locally homogeneous space SL(2,R) which are invariant and have positive entropy under the diagonal subgroup of SL(R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case.
Journal ArticleDOI
Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics
TL;DR: In this article, the authors review various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics, and adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution.
Journal ArticleDOI
Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors
TL;DR: In this paper, the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to non-hyperbolic systems, and the generalized partial dimensions of the invariant measure and the distribution of effective Liapunov exponents for hyperbolic attractors are investigated.
Journal ArticleDOI
Differentiation of SRB States
TL;DR: In this paper, the Onsager reciprocity relations and the fluctuation-dissipation formula of nonequilibrium statistical mechanics have been studied in hyperbolic manifold diffeomorphisms.
References
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Journal ArticleDOI
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
TL;DR: In this article, the authors present an agreement between Publications mathematiques de l'I.H.E.S. and les conditions generales d'utilisation (http://www.numdam.org/legal.php).
Journal ArticleDOI
Ergodic theory of differentiable dynamical systems
TL;DR: In this paper, the existence of stable manifolds is proved almost everywhere with respect to everyf-invariant probability measure on a compact manifold M. The proof of this stable manifold theorem is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products.
Book ChapterDOI
A measure associated with axiom-a attractors.
TL;DR: In this article, the future orbits of a diffeomorphism near an Axiom-A attrac-tor are investigated and it is found that their asymptotic behavior is in general described by a fixed probability measure yt carried by the attractor.
Journal ArticleDOI
An inequality for the entropy of differentiable maps
David Ruelle,David Ruelle +1 more