A natural boundary for the dynamical zeta function for commuting group automorphisms
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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.References
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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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3336617