On Flow Equivalence of the Subshifts Associated with the Stream Version of Asymmetric Binary Systems

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l$</tex> of a nonnegative integer and the probability parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$0 &lt; p &lt; 1$</tex>. The algorithm of stream encoding yields the edge shift <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{X}_{G}$</tex> associated with the stream version of ABS while the algorithm produces the edge shift <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{X}_{H}$</tex> associated with output blocks from the stream version of ABS. Previously, we have shown that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{X}_{G}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{X}_{H}$</tex> have the same topological entropy. In this research, we find that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{X}_{G}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{X}_{H}$</tex> are flow equivalent. We consider the case where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p=1/\beta$</tex> with the golden mean <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta=(1+\sqrt{5})/2$</tex>. For several <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l^{\prime}\mathbf{s}$</tex>, we compute the Parry-Sullivan quantity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{D}(\boldsymbol{A}_{G})$</tex> and the Bowen-Franks group <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{BF}(\boldsymbol{A}_{G})$</tex> since they form a complete set of flow equivalence invariants for irreducible shift of finite type. The results imply that for a fixed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p, \boldsymbol{D}(\boldsymbol{A}_{G})$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{BF}(\boldsymbol{A}_{G})$</tex> depend on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l$</tex>.