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October  2005, 13(5): 1203-1215. doi: 10.3934/dcds.2005.13.1203

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  March 2004 Revised  July 2005 Published  September 2005

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.
Keywords: Anosov systems., zeta functions, Dynamical determinants.
Mathematics Subject Classification: 37D20, 37C3.
Citation: Carlangelo Liverani. Fredholm determinants, Anosov maps and Ruelle resonances. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1203-1215. doi: 10.3934/dcds.2005.13.1203
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