Fredholm determinants, Anosov maps and Ruelle resonances

Carlangelo Liverani

doi:10.3934/dcds.2005.13.1203

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2005, Volume 13, Issue 5: 1203-1215. Doi: 10.3934/dcds.2005.13.1203

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Fredholm determinants, Anosov maps and Ruelle resonances

Carlangelo Liverani^1,

1.
Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, I-00133 Roma

Received: February 29, 2004

Revised: June 30, 2005

Published: September 2005

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Abstract

I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for $\C^\infty$ Anosov diffeomorphisms, the dynamical determinant is an entire function.

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Mathematics Subject Classification: 37D20, 37C30.

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