Talk by Vsevolod Chernyshev

I will discuss the following dynamical system on a metric graph with real lengths of edges. Let a point start its motion along one of the edges at the initial moment of time. The passage time for each individual edge is fixed. At each vertex of degree n, when k ≤ n points come simultaneously, n new moving points emerge. Backward turns on the edges areprohibited in this model. One could find asymptotics for the number N(T) of such moving points as the time T increases, i.e. number of all possible lengths of paths on metric graphs that do not exceed T. The consideration of such a dynamical system is motivated by the study of the evolution of narrow wave packets on undirected metric graphs and the study of multitilings for the case of directed graphs. Solutions to this problem, depending on the type of graph, are associated with different problems of number theory. An overview of the results, which depend on the arithmetic properties of lengths, will be given as well as review of open problems.