Ergodic Theory and Dynamical Systems



Zeta functions for elements of entropy rank-one actions


RICHARD MILES a1
a1 School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK (e-mail: r.miles@uea.ac.uk)

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Abstract

An algebraic $\mathbb{Z}^d$-action of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading to a uniform parameterization of the zeta functions that arise in expansive components of an expansive action, together with necessary and sufficient conditions for rationality in a more general setting.

(Published Online February 12 2007)
(Received January 19 2006)
(Revised August 2 2006)