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

A language L over a finite alphabet Sigma is growth sensitive (or entropy sensitive) if forbidding any finite set of factors F of L yields a sublanguage L-F 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 Sigma. Considering the graph as an (infinite) automaton, we associate with any pair of vertices x, y is an element of X the language L-x,L-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. (C) 2010 Elsevier B.V. All rights reserved.