A generalization of the simulation theorem for semidirect products

TL;DR

This paper extends the simulation theorem to semidirect products of groups, showing that effectively closed dynamical systems can be embedded into subshifts of finite type for these groups, with applications to aperiodic subshifts.

Contribution

It generalizes Hochman's result from $bZ^d$ actions to semidirect products of groups with $bZ^2$, broadening the scope of symbolic dynamics embeddings.

Findings

Realization of effectively closed $H$-systems as $G$-subshifts of finite type.

Existence of non-empty strongly aperiodic subshifts for groups with decidable word problem.

Extension of $H$-actions to $G$-sofic subshifts in the case of expansive actions.

Abstract

We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed $\mathbb{Z}^d$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^2$. Let $H$ be a finitely generated group and $G = \mathbb{Z}^2 \rtimes H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,f)$ where $Y$ is a Cantor set, there exists a $G$-subshift of finite type $(X,\sigma)$ such that the $H$-subaction of $(X,\sigma)$ is an extension of $(Y,f)$. In the case where $f$ is an expansive action of a recursively presented group $H$, a subshift conjugated to $(Y,f)$ can be obtained as the $H$-projective subdynamics of a $G$-sofic subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.