The homeomorphisms of the Sierpiński carpet are not classifiable by countable structures

TL;DR

This work proves that the group of homeomorphisms of the Sierpiński carpet, $(\mathcal{H}(\mathcal{S}), \approx)$, is not classifiable by countable structures. The authors reduce a known non-classifiable orbit equivalence problem for the group $G=\prod_{\mathbb{N}}\mathbb{Z}$ with $E_{\widehat{G}}$ to conjugacy of homeomorphisms of $\mathcal{S}$ via a Borel map $\rho$, and establish this reduction through two technical lemmas that provide controlled deballings and extensions. The core strategy bypasses the lack of linear structure on $\mathcal{S}$ by using deballing techniques and precise extensions with bounded distortion, yielding a turbulence-like obstruction in the 1-dimensional planar setting. The result highlights fundamental limits of classification by countable structures for low-dimensional continua and informs the broader program of turbulence-type obstructions to classifiability by homeomorphism groups. All mathematical notation is kept within $...$ delimiters to ensure precise representation of the key objects and relations.

Abstract

We show that the homeomorphisms of the Sierpiński carpet are not classifiable, up to conjugacy, using isomorphism types of countable structures as invariants.

Figures (6)

• Figure 1: The Sierpinski Carpet $\mathcal{S}$
• Figure 2:
• Figure 3: A dynamical realization of $\mathcal{S}$
• Figure 4: The sets $T, \widetilde{T}\subseteq L_n$ and $\sigma(T)\subseteq \widetilde{T}$, if $\sigma(n)=3$ and $\sigma(n-1)=4$.
• Figure 5: Decomposing $J$ into the trapezoids $T_{a,b}$

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 2.1: Whyburn1958
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Claim 3.3.1
• proof
• Claim 3.3.2
• proof
• proof : Proof of Lemma \ref{['L:main1']}