On entropy and intrinsic ergodicity of coded subshifts

Abstract

Any coded subshift X C X_C defined by a set C C of code words contains a subshift, which we call L C L_C , consisting of limits of single code words. We show that when C C satisfies the unique decipherability property, the topological entropy h ( X C ) h(X_C) of X C X_C is determined completely by h ( L C ) h(L_C) and the number of code words of each length. More specifically, we show that h ( X C ) = h ( L C ) h(X_C) = h(L_C) exactly when a certain infinite series is less than or equal to 1 1 , and when that series is greater than 1 1 , we give a formula for h ( X C ) h(X_C) . In the latter case, an immediate corollary (using a result from [Israel J. Math. 192 (2012), pp. 785–817] is that X C X_C has a unique measure of maximal entropy.