Linear response for Dirac observables of Anosov diffeomorphisms

Abstract : We consider a $\mathcal{C}^3$ family $t\mapsto f_t$ of $\mathcal{C}^4$ Anosov diffeomorphisms on a compact Riemannian manifold $M$. Denoting by $\rho_t$ the SRB measure of $f_t$, we prove that the map $t\mapsto\int \theta d\rho_t$ is differentiable if $\theta$ is of the form $\theta(x)=h(x)\delta(g(x)-a)$, with $\delta$ the Dirac distribution, $g:M\rightarrow \mathbb{R}$ a $\mathcal{C}^4$ function, $h:M\rightarrow\mathbb{R}$ a $\mathcal{C}^3$ function and $a$ a regular value of $g$. We also require a transversality condition, namely that the intersection of the support of $h$ with the level set $\{g(x)=a\} $ is foliated by 'admissible stable leaves'.
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Submitted on : Thursday, August 1, 2019 - 2:08:45 AM
Last modification on : Wednesday, August 14, 2019 - 11:21:34 PM

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Matthieu Porte. Linear response for Dirac observables of Anosov diffeomorphisms. Ergod.Theor.Dynam.Syst.A, 2019, 39, pp.1799-1819. ⟨10.3934/dcds.2019078⟩. ⟨hal-02222999⟩

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