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Banach spaces adapted to Anosov systems

Published online by Cambridge University Press:  08 December 2005

SÉBASTIEN GOUËZEL
Affiliation:
IRMAR, Université de Rennes 1, Campus de Beaulieu, bâtiment 22, 35042 Rennes Cedex, France (e-mail: sebastien.gouezel@univ-rennes1.fr)
CARLANGELO LIVERANI
Affiliation:
Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy (e-mail: liverani@mat.uniroma2.it)
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Abstract

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We study the spectral properties of the Ruelle–Perron–Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the ${\mathcal C}^\infty$ case, the essential spectral radius is arbitrarily small, which yields a description of the correlations with arbitrary precision. Moreover, we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the Sinai–Ruelle–Bowen measure, the variance for the central limit theorem, the rates of decay for smooth observable, etc.).

Type
Research Article
Copyright
2005 Cambridge University Press