A generalization of the simulation theorem for semidirect products
Résumé
We generalize a result of Hochman in two simultaneous directions: instead of realizing an arbitrary effectively closed Z^d action as a factor of a subaction of a Z^{d+2}-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with Z^2. Let H be a finitely generated group and G = Z^2 \times_φ H a semidirect product. We show that for any effectively closed H -dynamical system (Y, T ) where Y ⊂ {0, 1}^N, there exists a G-subshift of finite type (X, σ) such that the H-subaction of (X, σ) 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.