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DYNAMICAL ZETA FUNCTIONS FOR ANALYTIC SURFACE DIFFEOMORPHISMS WITH DOMINATED SPLITTING

Published online by Cambridge University Press:  08 March 2005

Viviane Baladi ,
Enrique R. Pujals  and
Martín Sambarino
Viviane Baladi
Affiliation:
CNRS UMR 7586, Institut Mathématique de Jussieu, 75251 Paris, France ([email protected])
Enrique R. Pujals
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil ([email protected]) IMPA, Estrada dona Castorina, CEP 22460-320, Rio de Janeiro, Brazil
Martín Sambarino
Affiliation:
IMERL, Facultad de Ingeniería, CC 30, 11300 Montevideo, Uruguay ([email protected])
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Abstract

We consider a real-analytic compact surface diffeomorphism $f$, for which the tangent space over the non-wandering set $\varOmega$ admits a dominated splitting. We study the dynamical determinant

$$ d_f(z)=\exp-\sum_{n\ge1}\frac{z^n}{n}\sum_{x\in\textrm{Fix}^*f^n}|\textrm{Det}(Df^n(x)-\textrm{Id})|^{-1}, $$

where $\textrm{Fix}^*f^n$ denotes the set of fixed points of $f^n$ with no zero Lyapunov exponents. By combining previous work of Pujals and Sambarino on $C^2$ surface diffeomorphisms with, on the one hand, results of Rugh on hyperbolic analytic maps and, on the other, our two-dimensional version of the same author’s analysis of one-dimensional analytic dynamics with neutral fixed points, we prove that $d_f(z)$ is either an entire function or a holomorphic function in a (possibly multiply) slit plane.

AMS 2000 Mathematics subject classification: Primary 37C30; 37D30; 37E30

Keywords

dynamical zeta functionsdominated splittingsurface diffeomorphismstransfer operators
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 4 , Issue 2 , April 2005 , pp. 175 - 218
Copyright
2005 Cambridge University Press

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