Séminaires : Séminaire des Thésards

Equipe(s) : doctorants,
Responsables :Sébastien Biebler, Vincent Dumoncel, Elba Garcia-Failde, Thiago Landim, Odylo Costa, Francesca Rizzo, Antoine Sedillot
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Description

Le séminaire des thésards est l'occasion pour les doctorants de présenter des résultats et des problématiques dignes d'intérêt devant un public de non-spécialistes. L'ambiance y est informelle ; poser des questions naïves est encouragé, et les questions moins naïves sont bienvenues dans la mesure où elles n'entravent pas le bon déroulement de l'exposé.

Un jeudi sur deux à 18h00, en alternance entre Jussieu et Sophie Germain.

Orateur(s) Alexander Adam - ,
Titre Resonances for Anosov diffeomorphisms
Date19/01/2017
Horaire18:00 à 19:00
Diffusion
Résume Deterministic chaotic behavior of invertible maps $T$ is appropriately described by the existence of expanding and contracting directions for the differential of $T$. A special class of such maps are Anosov diffeomorphisms. A famous example of such a diffeomorphism on the $2$-torus is induced by the matrix $M=\begin{pmatrix}2&1\\1&1\end{pmatrix}$. For all pairs of $L_2$-functions, real-analytic on the $2$-torus, one defines a correlation function for $T$ which captures the independence of such a pair under the evolution $T^n$ as $n\to\infty$. What is the rate of convergence of the correlation as $n\to\infty$, e.g. how fast is the mixing of $T$? The behavior of the correlation for $M$ is well-understood. In this talk I consider small perturbations $T$ of $M$. The composition operator $\mathcal{K}g:=g\circ T$, acting on a suitable Hilbert space, allows us to study the correlation from a functional analytic point of view. The eigenvalues of $\mathcal{K}$ are related to the zeros - the Ruelle resonances - of the analytically continued dynamical determinant of $T$. The trivial resonance is $1$ and it is the only one for $M$. The existence of a non-trivial resonance would change the speed of mixing drastically. For the considered perturbations at least one non-trivial resonance of $T$ appears.
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