We will study a certain synchronizing property of subshifts called $λ$-synchronization. The $λ$-synchronizing subshifts form a large class of irreducible subshifts containing irreducible sofic shifts. We prove that the $λ$-synchronization is invariant under flow equivalence of subshifts. The $λ$-synchronizing K-groups and the $λ$-synchronizing Bowen-Franks groups are studied and proved to be invariant under flow equivalence of $λ$-synchronizing subshifts. They are new flow equivalence invariants for $λ$-synchronizing subshifts.
A $λ$-graph system $\frak L$ is a labeled Bratteli diagram with shift operation. It is a generalized notion of finite labeled graph and presents a subshifts. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and $C^*$-algebras constructed from $λ$-graph systems.
We show that within any strong orbit equivalent class, there exist minimal subshifts with arbitrarily low superlinear complexity. This is done by proving that for any simple dimension group with unit $(G,G^+,u)$ and any sequence of positive numbers $(p_n)_{n\in\mathbb{N}}$ such that $\lim n/p_n=0$, there exist a minimal subshift whose dimension group is order isomorphic to $(G,G^+,u)$ and whose complexity function grows slower than $p_n$. As a consequence, we get that any Choquet simplex can be realized as the set of invariant measures of a minimal Toeplitz subshift whose complexity grows slow…
For a finite digraph $D$, we define the corresponding subshift of finite type $(X_D, σ_D)$ to be the dynamical system where $X_D$ is the set of all bi-infinite walks through $D$ and $σ_D$ is the shift operator. Two digraphs $D_1$ and $D_2$ are called shift equivalent if there is $m\geq 1$ such that $(X_{D_1}, σ_{D_1}^n)$ and $(X_{D_2}, σ_{D_2}^n)$ are topologically conjugate for all $n\geq m$. They are called strong shift equivalent if this holds for $m=1$. In this paper we introduce a new category which generalises the category of subshifts of finite type and topological conjugacy. Our catego…
It is shown that for a non-singular conservative shift on a topologically mixing Markov subshift with Doeblin Condition the only possible absolutely continuous shift-invariant measure is a Markov measure. Moreover, if it is not equivalent to a homogeneous Markov measure then the shift is of Krieger-type $\mathrm{III}_1$. A criterion for equivalence of Markov measures is included.
