Seminar

# Markov-Dyck shifts

#### Classification of operator algebras: complexity, rigidity, and dynamics

#### 10 March 14:00 - 15:00

#### Wolfgang Krieger - Heidelberg University

Markov-Dyck shifts are symbolic dynamical systems, that are constructed from directed graphs (always assumed finite and strongly connected), by splitting the edges of the graph into a "negative edge" , which is interpreted as an open bracket, and a "positive edge", which is interpreted as a closed bracket. The alphabet of the Markov-Dyck shift is the collection of "negative" and "positive" edges, and the admissible words of the Markov-Dyck shift are formed by imposing rules, that are familiar from the compatibility rules for open and closed brackets. The question is: Does the topological conjugacy of the Markov.Dyck shifts imply the isomorphism of the graphs, from which they were constructed? It is known that the answer is positive in the case of directed graphs, in which every vertex has at least two incoming edges. In the case, that there are vertices with only one incoming edge, the edges, that are the single incoming edges of their target vertices, form a collection of subtrees of the graph. These trees are rooted.We call them contractible subtrees. If they are contracted to their root, then the result is a directed graph, in which every vertex has at least two incoming edges. In this talk we consider the case that the directed graph has only one contracting subtree. We discuss two examples, in which the structure of the contacting subtree, and the structure of the directed graph, can be determined from dynamical data, like periodic orbit counts, of the Markov-Dyck shift. Therefore, in these examples the answer to the above question is also positive. This is joint work with Kengo Matsumoto.

Organizers

Marius Dadarlat

Purdue University

Søren Eilers

University of Copenhagen

Asger Törnquist

University of Copenhagen