A natural boundary for the dynamical zeta function for commuting group automorphisms

Miles, Richard

doi:10.1090/S0002-9939-2015-12515-4

A natural boundary for the dynamical zeta function for commuting group automorphisms
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Proc. Amer. Math. Soc. 143 (2015), 2927-2933 Request permission

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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