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 $${s \geq 1}$$ , for any integer $${\ell \geq 2}$$ , any sufficiently large r, any $${\varphi \in C^{r}(\mathbb{T}, \mathbb{R})}$$ such that the map $${f : \mathbb{T}^2 \to \mathbb{T}^2, f(x,y) = (\ell x, y + \varphi(x))}$$ is $${C^r}$$ -stably ergodic, there exists an open neighbourhood of f in $${C^r(\mathbb{T}^2,\mathbb{T}^2)}$$ 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 $${f: \mathbb{T}^3 \to \mathbb{T}^3}$$ such that all f′ in a C2 open neighbourhood of f possess a unique SRB measure $${\mu_{f'}}$$ and the map $${f' \mapsto \mu_{f'}}$$ is strictly Hölder at f, in particular, non-differentiable. This gives a partial answer to Dolgopyat’s Question 13.3 in Dolgopyat (Commun Math Phys 213:181–201, 2000).