Digit sets for connected tiles via similar matrices I: Dilation matrices with rational eigenvalues

TL;DR

This paper proves that for any m-dimensional dilation matrix with rational eigenvalues, there exists a digit set that produces a connected attractor, providing a practical condition for connectivity.

Contribution

It establishes the existence of digit sets ensuring connected attractors for dilation matrices with rational eigenvalues and introduces a sufficient condition for a specific digit set.

Findings

Existence of digit sets for connected attractors with rational eigenvalues

A practical condition for the centered canonical digit set to produce connected attractors

Theoretical framework for connected tiles in dilation systems

Abstract

Given any m-dimensional dilation matrix A with rational eigenvalues, we demonstrate the existence of a digit set D such that the attractor T(A,D) of the iterated function system generated by A and D is connected. We give an easily verified sufficient condition on A for a specific digit set, which we call the centered canonical digit set for A, to give rise to a connected attractor T(A,D).