Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-30T23:32:49.708Z Has data issue: false hasContentIssue false

DYNAMICAL ZETA FUNCTIONS FOR ANALYTIC SURFACE DIFFEOMORPHISMS WITH DOMINATED SPLITTING

Published online by Cambridge University Press:  08 March 2005

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])

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

Type
Research Article
Copyright
2005 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)