A dichotomy in orbit-growth for commuting automorphisms

TL;DR

This paper investigates the growth of periodic orbits in systems with commuting automorphisms, revealing a dichotomy where some systems have more or fewer orbits than expected based on entropy.

Contribution

It introduces a dichotomy in orbit-growth behavior for expansive actions by commuting automorphisms of compact groups.

Findings

Systems with excess periodic orbits beyond entropy predictions

Systems with no excess of periodic orbits

Identification of conditions leading to each behavior

Abstract

We consider asymptotic orbit-counting problems for certain expansive actions by commuting automorphisms of compact groups. A dichotomy is found between systems with asymptotically more periodic orbits than the topological entropy predicts, and those for which there is no excess of periodic orbits.