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Anosov diffeomorphisms and coupling

Published online by Cambridge University Press:  09 January 2002

XAVIER BRESSAUD  and
CARLANGELO LIVERANI
XAVIER BRESSAUD
Affiliation:
Instituto de Matematica e Estatistica, Universidade de São Paulo, Rua do Matão 1010, 05508-900 São Paulo SP, Brazil (e-mail: [email protected])
CARLANGELO LIVERANI
Affiliation:
Dipartimento di Matematica, Universita' di Roma ‘Tor Vergata’, via della ricerca Scientifica, 00133 Rome, Italy (e-mail: [email protected])
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Abstract

We prove the existence of a Sinai–Ruelle–Bowen (SRB) measure and the exponential decay of correlations for smooth observables for mixing Anosov C^{1+\alpha} diffeormorphisms on a d-dimensional (d \geq 2) Riemannian manifold. The novelty lies in the very simple method of proof. We construct explicitly a coupling between two initial densities so that under the action of the diffeomorphism, both components get closer and closer. The speed of this convergence can be explicitly estimated and is directly related to the speed of decay of the correlations. The existence of the SRB measure and its properties readily follow.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 1 , February 2002 , pp. 129 - 152
Copyright
2002 Cambridge University Press

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