Subsystems of Shifts of Finite Type over Countable Amenable Groups

This dissertation is primarily concerned with the subsystem problem for subshifts of finite type (SFTs) on countable amenable groups. Firstly, we demonstrate that an SFT with positive entropy exhibits a ubiquity of subsystems. We prove that for any countable amenable group G, if X is a G-SFT with positive topological entropy h(X) > 0 and Y ⊂ X is a subshift such that h(Y) < h(X), then the entropies of the SFTs Z which satisfy Y ⊂ Z ⊂ X are dense in the interval [h(Y), h(X)]. Secondly, we present an embedding theorem which provides conditions under which a given subshift may be realized as a subsystem of a given SFT. Let G be a countable amenable group with the comparison property. Let X be a strongly aperiodic subshift over G. Let Y be a strongly irreducible shift of finite type over G which has no global period, meaning that the shift action is faithful on Y. If h(X) < h(Y) and Y contains at least one factor of X, then X embeds into Y. Our proofs rely on recent developments in the theory of tilings and quasi-tilings of amenable groups.