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On the Smooth Dependence of SRB Measures for Partially Hyperbolic Systems

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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).

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  1. Institut de Mathématique de Jussieu—Paris Rive Gauche, Cedex 13, 75205, Paris, France

    Zhiyuan Zhang

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  1. Zhiyuan Zhang
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Communicated by C. Liverani

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Zhang, Z. On the Smooth Dependence of SRB Measures for Partially Hyperbolic Systems. Commun. Math. Phys. 358, 45–79 (2018). https://doi.org/10.1007/s00220-018-3088-x

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