Topological properties of factor maps in symbolic dynamics

Abstract

We study the topological and dynamical properties of sliding block codes between shift spaces together with the existence and extensions of these codes. Given a code from a shift space to an irreducible sofic shift, any two of the following three conditions ─ open, constant-to-one, (right or left) closing ─ imply the third. If the range is not sofic, then the same result holds when bi-closingness replaces closingness. Properties of open mappings between shift spaces are investigated in detail. In Chapter 3, properties of infinite-to-one codes are given. For an almost specified shift X, we show that an irreducible shift of finite type Y of lower entropy is a factor of X if and only if it is a factor of X by an open bi-continuing code. If these equivalent conditions hold and Y is mixing, then any code from a proper subshift of X to Y can be extended to an open bi-continuing code on X. In Chapter 4, we find a connection between symbolic dynamics and graph theory. Given two graphs G and H, there is a bi-resolving (or bi-covering) graph homomorphism from G to H if and only if their adjacency matrices satisfy certain matrix relations. We give several sufficient conditions for a bi-resolving homomorphism to have a bi-covering extension with an irreducible domain. Using these results, we prove that a bi-closing code between subshifts can be extended to an n-to-1 code between irreducible shifts of finite type for all large n.