Linear response for Dirac observables of Anosov diffeomorphisms

Matthieu Porte

doi:10.3934/dcds.2019078

Article Contents

2019, Volume 39, Issue 4: 1799-1819. Doi: 10.3934/dcds.2019078

This issue Previous Article Spectra of expanding maps on Besov spaces Next Article A non-local problem for the Fokker-Planck equation related to the Becker-Döring model

Linear response for Dirac observables of Anosov diffeomorphisms

Matthieu Porte^1,2,

1.
École normale supérieure, Département de mathématiques et applications, Paris, 75005, France
2.
Current affiliation, AgroParisTech, Département de sciences économiques, sociales et de gestion, Paris, 75015, France

Received: October 2017

Revised: September 2018

Published: January 2019

This work was conducted in 2017 as a MSc thesis at UPMC, under the direction of V. Baladi (CNRS/IMJ-PRG), whom the author thanks for her guidance and her useful remarks at all stages of this work. The author thanks P-A. Guihneuf (UMPC/IMJ-PRG) for his helpful comments. The author is also grateful to the reviewers for their insights on this paper throughout the drafting process

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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{C} $ 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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Mathematics Subject Classification: Primary: 37C40; Secondary: 37C30, 37D20.

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Figure 1. Behavior of the support of a Dirac observable on the stable manifold under the reverse dynamics. The two axes through the origin are the unstable and stable directions. The parallel lines are level sets for the function g(u, s) = u with the line passing through the origin being the level set {g = 0}. In the left side picture, the shaded area is the support of h. The right side picture is the image of the left side picture by f⁻¹. Both pictures are in $\mathbb{R}^2$, each of the grid squares corresponds to a fundamental domain of the torus.

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Figure 2. Behavior of the support of a Dirac observable with g(x, y) = y − 0.4x under the reverse dynamics. The stable and unstable directions are the same as befor. Parallel lines are level sets for g. The line through the origin is the level set {g = 0}. The shaded area is the support of h. The right side picture is the image of the left side one by f⁻¹.

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Figure 3. Behavior of the support of a Dirac observable with g(x, y) = x² + y² under the reverse dynamics. In the left side picture, circles are level sets of g and the middle circle is the level set {g = 0.4²}. The shaded area is the support of h. Note that it excludes the region where the middle circle is tangent to the unstable direction. The right side picture is the image of the left side one by f⁻¹.

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