Periodic points and residual finiteness of automorphism groups of subshifts

TL;DR

This paper investigates the relationship between the density of periodic points in subshifts and the residual finiteness of their automorphism groups, establishing new connections and counterexamples in symbolic dynamics.

Contribution

It proves that non-density of periodic points implies non-residual finiteness of automorphism groups for certain subshifts, and constructs examples with non-residually finite automorphism groups.

Findings

Density of periodic points implies residual finiteness of automorphism groups.

Existence of strongly irreducible $ extbf{Z}^2$-subshifts with non-residually finite automorphism groups.

Automorphism groups of block gluing $ extbf{Z}^2$-subshifts are locally embeddable in finite groups.

Abstract

If totally periodic points are dense in a subshift $X$, its automorphism group is residually finite. We show a weak converse: if periodic points are not dense in a subshift $X$, then the automorphism group of $X \times Y$ is not residually finite for full shifts $Y$ (and sufficiently full-shift-like subshifts). On the other hand, we show that the automorphism group of a block gluing $\Z^2$-subshift is always locally embeddable in finite groups (thus sofic). Hochman recently constructed a strongly irreducible $\Z^2$-subshift with no periodic points. Combining our result with this example gives a strongly irreducible $\Z^2$-subshift whose automorphism group is not residually finite, which solves a question of Coornaert and Ceccherini-Silberstein.