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The dynamical zeta function for commuting automorphisms of zero-dimensional groups

Published online by Cambridge University Press:  08 November 2016

RICHARD MILES  and
THOMAS WARD
RICHARD MILES
Affiliation:
MASH, University of Sheffield, Sheffield S10 2TN, UK email r.miles@sheffield.ac.uk
THOMAS WARD
Affiliation:
Level 13 Ziff Building, University of Leeds, LeedsLS2 9JT, UK email t.b.ward@leeds.ac.uk
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Abstract

For a $\mathbb{Z}^{d}$-action $\unicode[STIX]{x1D6FC}$ 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 $\unicode[STIX]{x1D6FC}$ 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 $\mathbb{Z}^{d}$. 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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 4 , June 2018 , pp. 1564 - 1587
Copyright
© Cambridge University Press, 2016 

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