Symmetric intersections of Rauzy fractals

TL;DR

This paper investigates symmetric subsets of Rauzy fractals formed by the intersection of fractals from a substitution and its reverse, revealing symmetry properties through the balanced pair algorithm.

Contribution

It introduces a method to analyze symmetric intersections of Rauzy fractals using the balanced pair algorithm for unimodular irreducible Pisot substitutions.

Findings

The intersection set is symmetric about the origin.

The balanced pair algorithm characterizes the intersection.

Symmetry is preserved under the reverse substitution.

Abstract

In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is reflection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions.