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A spectral gap for transfer operators of piecewise expanding maps

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


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