A generalization of the simulation theorem for semidirect products

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 $G={\mathbb{Z}}^2⋊H$ 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.

References in corpus (4)

• Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type
• Realization of aperiodic subshifts and uniform densities in groups
• A notion of effectiveness for subshifts on finitely generated groups
• Translation-like Actions and Aperiodic Subshifts on Groups

Cited by in corpus (4)

• Subsystem entropies of shifts of finite type and sofic shifts on countable amenable groups
• A geometric simulation theorem on direct products of finitely generated groups
• Arithmetical Hierarchy of the Besicovitch-Stability of Noisy Tilings
• Aperiodic subshifts of finite type on groups which are not finitely generated