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Spectral theory of ultradifferentiable hyperbolic dynamics

Abstract : In order to study the validity of a trace formula for infinitely differentiable Anosov flows proposed by Dyatlov and Zworski, we develop tools that allow us to investigate the Ruelle spectrum of infinitely differentiable hyperbolic dynamics. The notion of ultradifferentiability (and in particular the language of Denjoy-Carleman classes) plays a central role in our study. We give an ultradifferentiable analogue of the methods originally developed by Ruelle, Rugh and Fried (based on Grothendieck's results on nuclear operators) to study analytic hyperbolic dynamics. In particular, a detailed analysis of transfer operators associated to ultradifferentiable expanding maps of the circle is performed. The trace formula is then proved for a large class of ultradifferentiable Anosov flow. Finally, an analytic FBI transform is used in order to establish that the order of the dynamical determinant associated to an Anosov flow of Gevrey regularity is finite.
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Submitted on : Friday, July 2, 2021 - 4:07:34 PM
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  • HAL Id : tel-03096854, version 2

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Malo Jézéquel. Spectral theory of ultradifferentiable hyperbolic dynamics. Dynamical Systems [math.DS]. Sorbonne Université, 2020. English. ⟨NNT : 2020SORUS130⟩. ⟨tel-03096854v2⟩

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