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Zeta functions for certain non-hyperbolic systems and topological Markov approximations

Published online by Cambridge University Press:  01 December 1998

MICHIKO YURI
Affiliation:
Department of Business Administration, Sapporo University, Nishioka, Toyohira-ku, Sapporo 062, Japan

Abstract

We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps. In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition. We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, i.e. a characterization of the radius of convergence in terms of pressure. The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems. The Markov approximation method then gives a product formula of zeta functions for certain weighted functions.

Type
Research Article
Copyright
1998 Cambridge University Press

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