Entropy sensitivity of languages defined by infinite automata, via Markov chains with forbidden transitions

A language L over a finite alphabet is growth sensitive (or entropy sensitive) if forbidding any finite set of factors F of L yields a sublanguage LF whose exponential growth rate (entropy) is smaller than that of L. Let (X,E,ℓ) be an infinite, oriented, edge-labelled graph with label alphabet . Considering the graph as an (infinite) automaton, we associate with any pair of vertices x,y∈X the language Lx,y consisting of all words that can be read as labels along some path from x to y. Under suitable general assumptions, we prove that these languages are growth sensitive. This is based on using Markov chains with forbidden transitions.