Effect of quantified irreducibility on the computability of subshift entropy
Silvère Gangloff, Benjamin Hellouin de Ménibus · Discrete and Continuous Dynamical Systems · 2019
We study the algorithmic computability of topological entropy of subshifts subjected to a quantified version of a strong condition of mixing, called irreducibility. For subshifts of finite type, it is known that this problem goes from uncomputable to computable as the rate of irreducibility decreases. Furthermore, the set of possible values for the entropy goes from all right-recursively computable numbers to some subset of the computable numbers. However, the exact nature of the transition is not understood.In this text, we characterize a computability threshold for subshifts with decidable language (in any dimension), expressed as a summability condition on the rate function. This class includes subshifts of finite type under the threshold, and offers more flexibility for the constructions involved in the proof of uncomputability above the threshold. These constructions involve bounded density subshifts that control the density of particular symbols in all subwords.