Topology automaton and H\"older equivalence of Bara\'nski carpets

TL;DR

This paper extends the topology automaton method to Barański carpets, providing a broad criterion for their H"older and Lipschitz equivalence, thus advancing the classification of self-affine fractals.

Contribution

It introduces a topology automaton for Barański carpets and demonstrates its effectiveness in non-p.c.f. and self-affine contexts, generalizing previous approaches.

Findings

Established a general sufficient condition for H"older equivalence of Barański carpets.

Extended the topology automaton method to self-affine and non-p.c.f. fractals.

Provided tools for classifying Lipschitz and H"older equivalence in complex fractals.

Abstract

The study of Lipschitz equivalence of fractals is a very active topic in recent years. In 2023, Huang \emph{et al.} (\textit{Topology automaton of self-similar sets and its applications to metrical classifications}, Nonlinearity \textbf{36} (2023), 2541-2566.) studied the H\"older and Lipschitz equivalence of a class of p.c.f. self-similar sets which are not totally disconnected. The main tool they used is the so called topology automaton. In this paper, we define topology automaton for Bara\'nski carpets, and we show that the method used in Huang \emph{et al.} still works for the self-affine and non-p.c.f. settings. As an application, we obtain a very general sufficient condition for Bara\'nski carpets to be H\"older (or Lipschitz) equivalent.