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

TL;DR

This paper studies the dynamical zeta function for commuting automorphisms of zero-dimensional groups, proving the existence of natural boundaries in certain cases and analyzing the structure of periodic points.

Contribution

It proves Lind's conjecture on natural boundaries for the zeta function in specific classes of automorphism actions and explores the structure of periodic points in detail.

Findings

Natural boundary established for certain automorphism classes

Zero entropy actions have restricted periodic point structures

Supports Polya-Carlson dichotomy in dynamical zeta functions

Abstract

For a $\mathbb{Z}^d$-action $\alpha$ 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 $\alpha$ 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 Polya-Carlson dichotomy proposed by Bell and the authors.