Lipschitz equivalence of fractals and finite state automaton

TL;DR

This paper constructs a measure-preserving bi-Lipschitz map between two non-totally disconnected fractal squares using finite state automata, advancing understanding of Lipschitz equivalence beyond totally disconnected fractals.

Contribution

It introduces the first non-trivial bi-Lipschitz map between non-totally disconnected fractals, utilizing finite state automata.

Findings

Constructed a measure-preserving bi-Lipschitz map between fractal squares.

Extended Lipschitz equivalence concepts to non-totally disconnected fractals.

Demonstrated the applicability of automata in fractal geometry.

Abstract

The study of Lipschitz equivalence of fractals is a very active topic in recent years. Most of the studies in literature concern totally disconnected fractals. In this paper, using finite state automata, we construct a bi-Lipschitz map between two fractal squares which are not totally disconnected. This is the first non-trivial map of this type. We also show that this map is measure-preserving.