A dynamical zeta function for group actions

TL;DR

This paper develops a new dynamical zeta function for group actions, revealing its properties, connections to group zeta functions, and illustrating diverse behaviors through examples including shifts and virtually cyclic groups.

Contribution

It introduces a novel dynamical zeta function for group actions, establishes a product formula linking it to group zeta functions, and explores its properties across various group examples.

Findings

Full shifts can have rational zeta functions for infinitely many virtually cyclic groups.

For groups with Hirsch length ≥ 2, the zeta function typically has a natural boundary.

The zeta function relates to orbit growth and group dynamics.

Abstract

This article introduces and investigates the basic features of a dynamical zeta function for group actions, motivated by the classical dynamical zeta function of a single transformation. A product formula for the dynamical zeta function is established that highlights a crucial link between this function and the zeta function of the acting group. A variety of examples are explored, with a particular focus on full shifts and closely related variants. Amongst the examples, it is shown that there are infinitely many non-isomorphic virtually cyclic groups for which the full shift has a rational zeta function. In contrast, it is shown that when the acting group has Hirsch length at least 2, a dynamical zeta function with a natural boundary is more typical. The relevance of the dynamical zeta function in questions of orbit growth is also considered.