Generalized Oxtoby subshifts and hyperfiniteness

TL;DR

The paper addresses the problem of understanding the complexity of conjugacy for symbolic subshifts by linking it to the affine geometry of invariant-measure simplices. It develops a class of generalized Oxtoby subshifts built from Toeplitz sequences with a fixed period structure and shows that any compact metric Choquet simplex $C$ can be realized as the invariant-measure simplex $\\mathcal{M}_S(X)$ for some $X$ in this class. It then proves, via two distinct methods, that the conjugacy relation on this class is hyperfinite, using period-structure based reductions to finite block-permutations and an Oxtoby-property analysis. These results connect dynamical classification with hyperfinite equivalence relations, providing a tractable setting in which the invariant-measure geometry and conjugacy complexity align for a broad family of subshifts.

Abstract

We show that there exists a class of symbolic subshifts which realizes all Choquet simplices as simplices of invariant measures and the conjugacy relation on that class is hyperfinite.

Figures (2)

• Figure 1: Case 1
• Figure 2: Case 2

Theorems & Definitions (57)

• Theorem 1.1
• Definition 3.1
• Definition 3.2
• Definition 4.1
• Definition 4.2
• Theorem 4.3
• proof
• Definition 5.1
• Lemma 5.2
• proof