A natural boundary for the dynamical zeta function for commuting group automorphisms
Author:
Richard Miles
Journal:
Proc. Amer. Math. Soc. 143 (2015), 2927-2933
MSC (2010):
Primary 37A45, 37B05, 37C25, 37C30, 37C85, 22D40
DOI:
https://doi.org/10.1090/S0002-9939-2015-12515-4
Published electronically:
February 25, 2015
MathSciNet review:
3336617
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: For an action $\alpha$ of $\mathbb {Z}^d$ by homeomorphisms of a compact metric space, D. Lind introduced a dynamical zeta function and conjectured that this function has a natural boundary when $d\geqslant 2$. In this note, under the assumption that $\alpha$ is a mixing action by continuous automorphisms of a compact connected abelian group of finite topological dimension, it is shown that the upper growth rate of periodic points is zero and that the unit circle is a natural boundary for the dynamical zeta function.
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Additional Information
Richard Miles
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom
Email:
r.miles@uea.ac.uk
Received by editor(s):
September 19, 2013
Received by editor(s) in revised form:
January 9, 2014
Published electronically:
February 25, 2015
Communicated by:
Nimish Shah
Article copyright:
© Copyright 2015
American Mathematical Society
