Continuous hull of a combinatorial pentagonal tiling as an inverse limit

TL;DR

This paper constructs a topological space called the continuous hull for a regular pentagonal tiling, and shows how its associated dynamical system can be represented as an inverse limit with a shift map.

Contribution

It demonstrates how to represent the dynamical system of the continuous hull as an inverse limit, providing a new perspective on the tiling's combinatorial structure.

Findings

The continuous hull is a compact topological space.

The substitution map is a homeomorphism.

The dynamical system can be expressed as an inverse limit.

Abstract

In the article "Construction of the continuous hull for the combinatorics of a regular pentagonal tiling of the plane" we constructed a compact topological space for the combinatorics of "A regular pentagonal tiling of the plane", which we call the continuous hull. We also constructed a substitution map on the space which turns out to be a homeomorphism, and so the pair given by the continuous hull and the substitution map yields a dynamical system. In this paper we show how we can write this dynamical system as another dynamical system given by an inverse limit and a right shift map.