Scale function vs Topological entropy

TL;DR

This paper compares the scale function and topological entropy in topological automorphisms of totally disconnected locally compact groups, establishing inequalities, conditions for equality, and exploring their properties.

Contribution

It demonstrates that the logarithm of the scale function is always less than or equal to the topological entropy and provides conditions for equality, expanding understanding of these invariants.

Findings

Logarithm of scale function is dominated by topological entropy

Examples show the inequality can be strict

Conditions for equality between the invariants

Abstract

In the realm of topological automorphisms of totally disconnected locally compact groups, the scale function introduced by Willis in \cite{Willis} is compared with the topological entropy. We prove that the logarithm of the scale function is always dominated by the topological entropy and we provide examples showing that this inequality can be strict. Moreover, we give a condition equivalent to the equality between these two invariants. Various properties of the scale function, inspired by those of the topological entropy, are presented.