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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
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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

DOI: https://doi.org/10.1090/S0002-9939-2015-12515-4
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