This paper is a continuation of the paper entitled "Subshifts, λ-graph bisystems and C * -algebras", arXiv:1904.06464. A λ-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying certain compatibility condition on their edge labeling. For any two-sided subshift Λ, there exists a λ-graph bisystem satisfying a special property called FPCC. We will construct an AF-algebra F L with shift automorphism ρ L from a λ-graph bisystem (L − , L + ), and define a C * -algebra R L by the crossed product F L ⋊ ρ L Z. It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If λ-graph bisystems come from two-sided subshifts, these C * -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We will present a simplicity condition of the C * -algebra R L and the K-theory formulas of the C * -algebras F L and R L . The K-group for the AF-algebra F L is regarded as a two-sided extension of the dimension group of subshifts.Mathematics Subject Classification: Primary 37B10, 46L55; Secondary 46L35.

We study KMS states for gauge actions with potential functions on Cuntz-Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.

We prove that one-sided topological Markov shifts (X A , σ A ) and (X B , σ B ) for matrices A and B with entries in {0, 1} are continuously orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras ᏻ A and ᏻ B keeping their commutative C * -subalgebras C(X A ) and C(X B ). The "if" part (and hence the "only if" part) above is equivalent to the condition that there exists a homeomorphism from X A to X B intertwining their topological full groups. We will also study structure of the automorphisms of ᏻ A preserving the commutative C * -algebra C(X A ).

We point out incorrect lemmas in some papers regarding the C * -algebras associated with subshifts written by the second named author. To recover the incorrect lemmas and the affected main results, we will describe an alternative construction of C * -algebras associated with subshifts. The resulting C * -algebras are generally different from the originally constructed C * -algebras associated with subshifts and they fit the mentioned papers including the incorrect results. The simplicity conditions and the K-theory formulae for the originally constructed C * -algebras are described. We also introduce a condition called ( * ) for subshifts such that under this condition the new C * -algebras and the original C * -algebras are canonically isomorphic to each other. We finally present a subshift for which the two kinds of algebras have different K-theory groups.

Given a real number $\beta > 1$, we construct a simple purely infinite $C^*$-algebra ${\cal O}_{\beta}$ as a $C^*$-algebra arising from the $\beta$-subshift in the symbolic dynamics. The $C^*$-algebras $\{{\cal O}_{\beta} \}_{1<\beta \in {\Bbb R}}$ interpolate between the Cuntz algebras $\{{\cal O}_n\}_{1 < n \in {\Bbb N}}$. The K-groups for the $C^*$-algebras ${\cal O}_{\beta}$, $1 < \beta \in {\Bbb R}$, are computed so that they are completely classified up to isomorphism. We prove that the KMS-state for the gauge action on ${\cal O}_{\beta}$ is unique at the inverse temperature $\log \beta$, which is the topological entropy for the $\beta$-shift. Moreover, ${\cal O}_{\beta}$ is realized to be a universal $C^*$-algebra generated by $n-1=[\beta]$ isometries and one partial isometry with mutually orthogonal ranges and a certain relation coming from the sequence of $\beta$-expansion of $1$.