Differentiation of SRB states for hyperbolic flows

On Differentiability of Srb States for Partially Hyperbolic Systems

Consider a one parameter family of diffeomorphisms fε such that f0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let νε be any u-Gibbs state for fε. We prove (Theorem 1) that if A is a C ∞ function then the map A → νε(A) is differentiable at ε = 0. This implies (Corollary 1) that the difference of Birkhoff averages of the perturbed and un...

SRB measures for certain almost hyperbolic diffeomorphisms

Generalized Physical and Srb Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hénon maps. For this class of systems we are able to develop a complete theory about th...

Srb Measures of Certain Almost Hyperbolic Diffeomorphisms with a Tangency

We study topological and ergodic properties of some almost hyperbolic diffeomorphisms on two dimensional manifolds. Under generic conditions, diffeomorphisms obtained from Anosov by an isotopy pushing together the stable and unstable manifolds to be tangent at a fixed point, are conjugate to Anosov. For a finite codimension subset at the boundary of Anosov there exist a SRB measure and an uniqu...

Derivative Formulas for Generalized Srb Measure, Entropy, and Hausdorff Dimension of Hyperbolic Systems of Codimension One

Under the condition that unstable manifolds are one dimensional, the derivative formula of the potential function of the generalized SRB measure with respect to the underlying dynamical system is extended from the hyperbolic attractor case to the general case when the hyperbolic set intersecting with unstable manifolds is a Cantor set. It leads to derivative formulas of objects and quantities t...