Zd Symbolic Dynamics: Coding with an Entropy Inequality

Abstract

In this paper we discuss subsystem and coding results in Zd symbolic dynamics for d greater than 1. We prove that any Zd shift of finite type with positive topological entropy has a family of subsystems of finite type whose entropies are dense in the interval from zero to the entropy of the original shift. We show a similar result for Zd sofic shifts, and also show every Zd sofic shift can be covered by a Zd shift of finite type arbitrarily close in entropy. We also show that if a Z² shift of finite type with entropy greater than log N satisfies a certain mixing condition, then it must factor onto the full N-shift.