On the smooth dependence of SRB measures for partially hyperbolic systems

Abstract

In this paper, we study the differentiability of SRB measures for partially hyperbolic systems. We show that for any $s\ge 1$, for any integer $\ell \ge 2$, any sufficiently large $r$, any $φ\in C^{r}(\T, \R)$ such that the map $f : \T^2 \to \T^2, f(x,y) = (\ell x, y + φ(x))$ is $C^r-$stably ergodic, there exists an open neighbourhood of $f$ in $C^r(\T^2,\T^2)$ such that any map in this neighbourhood has a unique SRB measure with $C^{s-1}$ density, which depends on the dynamics in a $C^s$ fashion. We also construct a $C^{\infty }$ mostly contracting partially hyperbolic diffeomorphism $f: \T^3 \to \T^3$ such that all $f^{\prime }$ in a $C^2$ open neighbourhood of $f$ possess a unique SRB measure ${\mu }_{f^{\prime }}$ and the map $f^{\prime }\mapsto {\mu }_{f^{\prime }}$ is strictly Hölder at $f$, in particular, non-differentiable. This gives a partial answer to Dolgopyat's Question 13.3 in \cite{Do1}.

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References in corpus (4)

• Linear response for intermittent maps
• Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables
• Central limit theorem for the modulus of continuity of averages of observables on transversal families of piecewise expanding unimodal maps
• Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products

Cited by in corpus (2)

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• A criterion for transitivity of area preserving partially hyperbolic endomorphism on torus