Dynamical entropy of generalized quantum Markov chains on gauge invariant $C^*$-algebras

Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz-Krieger algebras

We introduce a notion of uniformly continuous orbit equivalence as a subequivalence relation of continuous orbit equivalence of one-sided topological Markov shifts. It is described in terms of gauge actions on the associated Cuntz-Krieger algebras and continuous full groups of the Markov shifts.

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

We study a class of ergodic quantum Markov semigroups on finite-dimensional unital $C^*$-algebras. These semigroups have a unique stationary state $σ$, and we are concerned with those that satisfy a quantum detailed balance condition with respect to $σ$. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to $σ$ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the $2$-Wasserstein metric, and in several interesting cases we are able to show, in an…

单侧拓扑马尔可夫移位连续轨道等价中的拓扑熵

Topological entropy in continuous orbit equivalence of one-sided topological Markov shifts

本文证明单侧拓扑马尔可夫移位的连续轨道等价类包含拓扑熵可任意大于或小于给定正实数的移位。

Topological entropy for countable Markov shifts and Exel--Laca algebras

We show that the (Gurevich) topological entropy for the countable Markov shift associated with an infinite transition matrix $A$ coincides with the non-commutative topological entropy for the Exel--Laca algebra associated with $A$, under certain conditions on $A$. An important example satisfying the conditions is the renewal shift, which is not locally finite. We also pose interesting questions for future research on non-commutative topological entropy for non-locally finite transition matrices.

Zeta functions and topological entropy of the Markov-Dyck shifts

The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller (G. Keller, {\it Circular codes, loop counting, and zeta-functions}, J. Combinatorial Theory {\bf 56} (1991), pp. 75--83). For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy ae determined.

On one-sided topological conjugacy of topological Markov shifts and gauge actions on Cuntz--Krieger algebras

We will characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz--Krieger algebras and its gauge actions with potentials.