Characterization and topological behavior of homomorphism tree-shifts

The purpose of this article is twofold. On one hand, we reveal the equivalence of shift of finite type between a one-sided shift X and its associated hom tree-shift T_X, as well as the equivalence in the sofic shift. On the other hand, we investigate the interrelationship among the comparable mixing properties on tree-shifts as those on multidimensional shift spaces. They include irreducibility, topologically mixing, block gluing, and strong irreducibility, all of which are defined in the spirit of classical multidimensional shift, complete prefix code (CPC), and uniform CPC. In summary, the mixing properties defined in all three manners coincide for T_X. Furthermore, an equivalence between irreducibility on T_A and irreducibility on X_A are seen, and so is one between topologically mixing on T_A and mixing property on X_A, where X_A is the one-sided shift space induced by the matrix A and T_A is the associated tree-shift. These equivalences are consistent with the mixing properties on X or X_A when viewed as a degenerate tree-shift.