# Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables

Baladi, V., Kuna, T. and Lucarini, V. (2017) Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables. Nonlinearity, 30 (3). 1204. ISSN 1361-6544

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To link to this item DOI: 10.1088/1361-6544/aa5b13

## Abstract/Summary

We consider a smooth one-parameter family $t\mapsto (f_t:M\to M)$ of diffeomorphisms with compact transitive Axiom A attractors $\Lambda_t$, denoting by $d \rho_t$ the SRB measure of $f_t|_{\Lambda_t}$. Our first result is that for any function $\theta$ in the Sobolev space $H^r_p(M)$, with $1<p<\infty$ and $0<r<1/p$, the map $t\mapsto \int \theta\, d\rho_t$ is $\alpha$-H\"older continuous for all $\alpha <r$. This applies to $\theta(x)=h(x)\Theta(g(x)-a)$ (for all $\alpha <1$) for $h$ and $g$ smooth and $\Theta$ the Heaviside function, if $a$ is not a critical value of $g$. Our second result says that for any such function $\theta(x)=h(x)\Theta(g(x)-a)$ so that in addition the intersection of $\{ x\mid g(x)=a\}$ with the support of $h$ is foliated by admissible stable leaves'' of $f_t$, the map $t\mapsto \int \theta\, d\rho_t$ is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables $\theta$ is motivated by extreme-value theory.

Item Type: Article Yes Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and StatisticsInterdisciplinary Research Centres (IDRCs) > Centre for the Mathematics of Planet Earth (CMPE) 69164 Institute of Physics