A spectral gap for transfer operators of piecewise expanding maps

Damien Thomine

doi:10.3934/dcds.2011.30.917

Article Contents

2011, Volume 30, Issue 3: 917-944. Doi: 10.3934/dcds.2011.30.917

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A spectral gap for transfer operators of piecewise expanding maps

Damien Thomine^1,

1.
45 rue d'Ulm, 75005 Paris

Received: April 30, 2010

Revised: October 31, 2010

Published: August 2011

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Abstract

We consider piecewise $\C^{1+\alpha}$ uniformly expanding maps on a Riemannian manifold, and study their invariant physical measures. We study the Perron-Frobenius operator on Sobolev spaces and bounded variation spaces, and prove that it is quasicompact if some conditions on the Lyapunov exponent and the combinatorial complexities are satisfied. Then, we get strong results concerning the existence of physical ergodic measures, and the exponential mixing of smooth observables.

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Mathematics Subject Classification: Primary: 37A25; Secondary: 47A35.

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