# Relative topological entropy for actions of non-discrete groups on compact spaces in the context of cut and project schemes

## TL;DR

This paper extends the theory of topological entropy to actions of non-discrete groups in aperiodic order, establishing key formulas and invariance properties crucial for dynamical analysis in cut and project schemes.

## Contribution

It generalizes Bowen's formula and invariance of topological entropy to non-discrete group actions, validating the Ornstein-Weiss lemma in this context.

## Key findings

- Bowen's formula extended to non-discrete groups
- Invariance of entropy from Van Hove sequence choice proven
- Ornstein-Weiss lemma validated for groups in cut and project schemes

## Abstract

In the study of aperiodic order via dynamical methods, topological entropy is an important concept. In this paper, parts of the theory, like Bowen's formula for fibre wise entropy or the independence of the definition from the choice of a Van Hove sequence, are extended to actions of several non-discrete groups. To establish these results, we will show that the Ornstein-Weiss lemma is valid for all considered groups which appear in the study of cut and project schemes.