On the topological entropy of subshifts of finite type on free semigroups

Abstract

In this paper, we provide an effective method to compute the topological entropies of GG-subshifts of finite type (GG-SFTs) with G=FdG=Fd and SdSd, the free group and free semigroup with dd generators respectively. We develop the entropy formula by analyzing the corresponding systems of nonlinear recursive equations (SNREs). Four types of SNREs of S2S2-SFTs, namely the types EE, DD, CC and OO, are introduced, and we could compute their entropies explicitly. This enables us to give the complete characterization of S2S2-SFTs on two symbols. That is, the set of entropies of S2S2-SFTs on two symbols is equal to E∪D∪C∪OE∪D∪C∪O. The methods developed in SdSd-SFTs will also apply to the study of the entropy theory of FdFd-SFTs. The entropy formulae of SdSd-, FdFd-golden mean shifts and kk-colored chessboards are also presented herein.