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Transfer operators for ultradifferentiable expanding maps of the circle

Part of: Smooth dynamical systems: general theory

Published online by Cambridge University Press:  11 June 2020

MALO JÉZÉQUEL
MALO JÉZÉQUEL*
Affiliation:
Laboratoire de Probabilités, Statistique et Modélisation (LPSM), CNRS, Sorbonne Université, Université de Paris, 4, Place Jussieu, 75005Paris, France email jezequel@lpsm.paris
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Abstract

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$, providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$.

Keywords

transfer operatordynamical determinantRuelle resonancesDenjoy–Carleman classes

MSC classification

Primary: 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2049 - 2068
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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