Asymptotic estimate for the counting problems corresponding to the dynamical system on some decorated graphs

Abstract

We study a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds, that is, a decorated graph. We consider the following dynamical system on decorated graphs. Suppose that, at the initial time, we have a narrow wave packet on a one-dimensional edge. It can be thought of as a point moving along the edge. When a packet arrives at the point of gluing, the expanding wavefront begins to spread on the Riemannian manifold. At the same time, there is a partial reflection of the wave packet. When the wavefront that propagates on the surface comes to another point of gluing, it generates a reflected wavefront and a wave packet on an edge. We study the number of Gaussian packets, that is, moving points on one-dimensional edges as time goes to infinity. We prove the asymptotic estimations for this number for the following decorated graphs: a cylinder with an interval, a two-dimensional torus with an interval and a three-dimensional torus with an interval. Also we prove a general theorem about a manifold with an interval and apply it to the case of a uniformly secure manifold.