Miles, R and Ward, T orcid.org/0000-0002-8253-5767 (2018) The dynamical zeta function for commuting automorphisms of zero-dimensional groups. Ergodic Theory and Dynamical Systems, 38 (4). pp. 1564-1587. ISSN 0143-3857
Abstract
For a Zd-action α by commuting homeomorphisms of a compact metric space, Lind introduced a dynamical zeta function that generalizes the dynamical zeta function of a single transformation. In this article, we investigate this function when α is generated by continuous automorphisms of a compact abelian zero-dimensional group. We address Lind’s conjecture concerning the existence of a natural boundary for the zeta function and prove this for two significant classes of actions, including both zero entropy and positive entropy examples. The finer structure of the periodic point counting function is also examined and, in the zero entropy case, we show how this may be severely restricted for subgroups of prime index in Zd. We also consider a related open problem concerning the appearance of a natural boundary for the dynamical zeta function of a single automorphism, giving further weight to the Pólya–Carlson dichotomy proposed by Bell and the authors.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016, Cambridge University Press. This article has been published in a revised form in Ergodic Theory and Dynamical Systems https://doi.org/10.1017/etds.2016.77. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. Uploaded in accordance with the publisher's self-archiving policy. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Chemistry (Leeds) > Physical Chemistry (Leeds) The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 21 Sep 2016 11:45 |
Last Modified: | 10 May 2018 15:13 |
Status: | Published |
Publisher: | Cambridge University Press |
Identification Number: | 10.1017/etds.2016.77 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:104961 |