This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle, and Bowen in the 1970s. SRB measures, as these objects are called, play an important role in the ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly speaking,

What Are SRB Measures, and Which Dynamical Systems Have Them?

We classify measures on the locally homogeneous space ?i\ SL(2,R) ?~ L which are invariant and have positive entropy under the diagonal subgroup of SL(2,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. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in theproof of the main result.

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Invariant measures and arithmetic quantum unique ergodicity

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics We 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 The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures In terms of these one can prove the Gallavotti–Cohen fluctuation theorem One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations At equilibrium one recovers in particular the Onsager reciprocity relations Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss The results just mentioned hold so far only for classical systems; they do not involve large size, ie, they hold without a thermodynamic limit

Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions For nonhyperbolic attractors, there are important modifications We discuss them for the examples of the logistic and Henon map We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics

Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

:Let f be a diffeomorphism of a manifold M, and ρf a (generalized) SRB state for f. If supp ρf is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup, and we compute the derivative. When supp ρf is an attractor, the derivative is given by
 
where X is the vector field . This formula can be extended to time dependent situations and also, at least formally, to nonuniformly hyperbolic situations.
 The above results will find their use in the study of the Onsager reciprocity relations and the fluctuation-dissipation formula of nonequilibrium statistical mechanics.

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Differentiation of SRB States

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 invariantes

https://www.jstor.org/stable/info/1971328

The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin's entropy formula

On considere f: (M,m)→(M,m) un C 2 -diffeomorphisme f d'une variete de Riemann compacte M preservant une mesure de probabilite de Borel m. Soit hm(f) l'entropie metrique de f et λ 1 (x)>...>λ r(x) (x) les exposants de Lyapunov distincts en x. On etablit une formule qui relie h m (f) et les λ r(x) (x)

François Ledrappier

Papers

The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin's entropy formula

The metric entropy of diffeomorphisms Part II: Relations between entropy, exponents and dimension

Positivity of the exponent for stationary sequences of matrices