An example of PET. Computation of the Hausdorff dimension of the aperiodic set

TL;DR

This paper introduces a new family of piecewise isometries, establishes a renormalization scheme, and computes the Hausdorff dimension of the discontinuity set using cocycles and continued fractions.

Contribution

It presents a novel family of piecewise isometries with a renormalization scheme and provides a method to compute the Hausdorff dimension of the discontinuity set.

Findings

Existence of a renormalization scheme within the family

Explicit computation of the Hausdorff dimension of the discontinuity set

Application of cocycles and continued fraction algorithms

Abstract

We introduce a family of piecewise isometries. This family is similar to the ones studied by Hooper and Schwartz. We prove that a renormalization scheme exists inside this family and compute the Hausdorff dimension of the discontinuity set. The methods use some cocycles and a continued fraction algorithm.