On differentiability of SRB states for partially hyperbolic systems

Abstract

Abstract. Consider a one parameter family of diffeomorphisms f ε such that f 0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let ν ε be any u-Gibbs state for f ε . We prove (Theorem 1) that if A is a C ∞ function then the map A → ν ε (A) is differentiable at ε = 0. This implies (Corollary 1) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to ε. We apply this result (Corollary 3) to show that if f 0 is a time one map of geodesic flow on a unit tangent bundle over a surface of negative curvature then a generic perturbation has a unique SRB measure with good statistical properties.

On the ergodicity of partially hyperbolic systems

“…Brin and Pesin's theorem applies in particular to the time-1 map ' 1 of the geodesic flow for a compact surface of constant negative curvature. If we make a C 1 small perturbation to ' 1 , all of their hypotheses continue to hold, except Lipschitzness of ᐃ c (this fact follows from combining results in [Dol04] with an argument of Mañé -see [BPSW01,p. 352…”

Stable ergodicity for partially hyperbolic attractors with negative central exponents

“…This statement is proven in [10] for the case when L is the whole manifold M , but the argument works in the general case as well.…”

“…In [11] a one-parameter family f ε is considered where f 0 is the time-1 map of the geodesic flow on the unit tangent bundle over a negatively curved surface. It is shown that in the volume preserving case, generically, either f ε or f −1 ε has negative central exponent for small ε and that there is an open set of nonconservative families where the central exponent is negative for any u-measure.…”

“…Given diffeomorphisms f and g, let F n = f n • g. It is shown in [11] that if f is either a T 1 extension of an Anosov diffeomorphism or the time-1 map of an Anosov flow and g is close to id, then for typical g and any sufficiently large n, either F n or F −1 n has negative central exponent with respect to any u-measure.…”

Linear response and periodic points

“…Contributions to the study of this and related problems have been made by Ruelle [15] and [16], Baladi and Smania [3], [4], [5]; Dolgopyat [9], Liverani and Butterley [6].…”