On Flow Equivalence of the Subshifts Associated with the Stream Version of Asymmetric Binary Systems
Hiroshi FUJISAKI · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences · 2024
The stream version of asymmetric binary systems (ABS) developed by Duda is an entropy coder for information sources with a finite alphabet. It has the state parameter$l$of a nonnegative integer and the probability parameter$p$with$0 < p < 1$. The algorithm of stream encoding yields the edge shift$\boldsymbol{X}_{G}$associated with the stream version of ABS while the algorithm produces the edge shift$\boldsymbol{X}_{H}$associated with output blocks from the stream version of ABS. Previously, we have shown that$\boldsymbol{X}_{G}$and$\boldsymbol{X}_{H}$have the same topological entropy. In this research, we find that$\boldsymbol{X}_{G}$and$\boldsymbol{X}_{H}$are flow equivalent. We consider the case where$p=1/\beta$with the golden mean$\beta=(1+\sqrt{5})/2$. For several$l^{\prime}\mathbf{s}$, we compute the Parry-Sullivan quantity$\boldsymbol{D}(\boldsymbol{A}_{G})$and the Bowen-Franks group$\boldsymbol{BF}(\boldsymbol{A}_{G})$since they form a complete set of flow equivalence invariants for irreducible shift of finite type. The results imply that for a fixed$p, \boldsymbol{D}(\boldsymbol{A}_{G})$and$\boldsymbol{BF}(\boldsymbol{A}_{G})$depend on$l$.