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

In this paper, we provide an effective method to compute the topological entropies of $G$-subshifts of finite type ($G$-SFTs) with $G=F_{d}$ and $S_{d}$, the free group and free semigroup with $d$ generators respectively. We develop the entropy formula by analyzing the corresponding systems of nonlinear recursive equations (SNREs). Four types of SNREs of $S_{2}$-SFTs, namely the types $\mathbf{E},\mathbf{D},\mathbf{C}$ and $\mathbf{O}$, are introduced and we could compute their entropies explicitly. This enables us to give the complete characterization of $S_{2}$-SFTs on two symbols. That is, the set of entropies of $S_{2}$-SFTs on two symbols is equal to $\mathbf{E}\cup \mathbf{D}\cup \mathbf{C}\cup \mathbf{O}$. The methods developed in $S_{d}$-SFTs will also be applied to the study of the entropy theory of $F_{d}$-SFTs. The entropy formulae of $S_{d}$-, $F_{d}$-golden mean shifts and $k$-colored chessboards are also presented herein.