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Topology automaton and Hölder equivalence of Barański carpets

Yunjie Zhu, Liang-yi Huang, chunbo Cheng

TL;DR

This paper extends the topology automaton framework to Barański carpets, a broad class of self-affine fractals, to obtain general sufficient conditions for Hölder and Lipschitz equivalence. By developing the cross automaton and its one-step simplifications, the authors connect symbolic dynamics with geometric structure, enabling a block-wise equivalence criterion based on horizontal blocks and their pairings. The introduction of a universal symbolic map g links different automata in a way that preserves surviving-time structure up to bounded distortion, which in turn yields isometries between the induced pseudo-metric spaces and, hence, Hölder (and Lipschitz for fractal squares) equivalence of carpets under the stated conditions. Overall, the work broadens the applicability of topology automata beyond p.c.f. self-similar sets to non-totally disconnected self-affine carpets, providing a practical criterion for classification and a blueprint for future generalizations.

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 Hölder 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 Barański 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 Barański carpets to be Hölder (or Lipschitz) equivalent.

Topology automaton and Hölder equivalence of Barański carpets

TL;DR

This paper extends the topology automaton framework to Barański carpets, a broad class of self-affine fractals, to obtain general sufficient conditions for Hölder and Lipschitz equivalence. By developing the cross automaton and its one-step simplifications, the authors connect symbolic dynamics with geometric structure, enabling a block-wise equivalence criterion based on horizontal blocks and their pairings. The introduction of a universal symbolic map g links different automata in a way that preserves surviving-time structure up to bounded distortion, which in turn yields isometries between the induced pseudo-metric spaces and, hence, Hölder (and Lipschitz for fractal squares) equivalence of carpets under the stated conditions. Overall, the work broadens the applicability of topology automata beyond p.c.f. self-similar sets to non-totally disconnected self-affine carpets, providing a practical criterion for classification and a blueprint for future generalizations.

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 Hölder 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 Barański 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 Barański carpets to be Hölder (or Lipschitz) equivalent.

Paper Structure

This paper contains 22 sections, 25 theorems, 79 equations, 7 figures.

Table of Contents

  1. Introduction
  2. $\Sigma$-automaton and pseudo-metric space
  3. Pseudo-metric space
  4. $\Sigma$-automaton
  5. Surviving time
  6. Pseudo-metric space induced by $\Sigma$-automaton
  7. Topology automaton of Barański carpet
  8. Companion IFS
  9. Topology automaton
  10. Coding and projection
  11. Symmetric $\Sigma$-automaton
  12. Cross automaton
  13. Graph representation of cross automaton
  14. Feasibility of cross automaton
  15. Simplification of cross automaton
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $E\in {\mathcal{B}}(n, m_1,{\mathcal{D}}_E)$ and $F\in {\mathcal{B}}(n, m_2,{\mathcal{D}}_F)$ be two Barański carpets satisfying the cross intersection condition, and assume that $E$, and also $F$, satisfies either the top isolated condition or the vertical separation condition. If there is a b

Figures (7)

  • Figure 1: (a) $E=K(5,\mathcal{D}_E)$ is a fractal square which satisfies the top isolated condition. (b) $F=K(5, 7, \mathcal{D}_F)$ is a Barański carpet which satisfies the vertical separation condition.
  • Figure 2: $K_1\simeq K_2$ and $K_3\simeq K_4$.
  • Figure 3: Some fractal squares with $n=3$, $\#{\mathcal{D}}=5$.
  • Figure 4: Illustration of proof of Theorem \ref{['thm:NA']}: Case 1.
  • Figure 5: The transition diagram of a cross automaton.
  • ...and 2 more figures

Theorems & Definitions (61)

  • Remark 1.1
  • Definition 1.1
  • Theorem 1.1
  • Example 1.1
  • Example 1.2
  • Example 1.3: Lipschitz classification of fractal squares with expanding factor $3$
  • Remark 1.2
  • Definition 2.1
  • Theorem 2.1: HWYZ23
  • Theorem 2.2: HWYZ23
  • ...and 51 more