Optimal parametrizations of a class of self-affine sets

TL;DR

This paper investigates the optimal parametrizations of invariant sets generated by single matrix graph iterated function systems, extending previous results and providing a foundation for space-filling curves of self-affine sets.

Contribution

It establishes that invariant sets of linear single matrix GIFS with primitive matrices and open set condition admit optimal parametrizations, generalizing prior work.

Findings

Invariant sets admit optimal parametrizations.

Results extend previous work on self-affine sets.

Foundation for space-filling curves of self-affine sets.

Abstract

In this paper, we study optimal parametrizations of the invariant sets of a single matrix graph IFS which is a generalization of the result of Rao and Zhang (2016). We show that the invariant sets of a linear single matrix GIFS which has a primitive associated matrix and satisfies the open set condition admit optimal parametrizations. This result is the basis of the further study of space-filling curves of self-affine sets.