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A generalization of the simulation theorem for semidirect products
Published online by Cambridge University Press: 10 April 2018
Abstract
We generalize a result of Hochman in two simultaneous directions: instead of realizing an arbitrary 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
$G=\mathbb{Z}^{2}\rtimes _{\unicode[STIX]{x1D711}}H$ a semidirect product. We show that for any effectively closed
$H$-dynamical system
$(Y,T)$ where
$Y\subset \{0,1\}^{\mathbb{N}}$, there exists a
$G$-subshift of finite type
$(X,\unicode[STIX]{x1D70E})$ such that the
$H$-subaction of
$(X,\unicode[STIX]{x1D70E})$ is an extension of
$(Y,T)$. In the case where
$T$ is an expansive action, a subshift conjugated to
$(Y,T)$ can be obtained as the
$H$-projective subdynamics of a sofic
$G$-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.
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