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

Published online by Cambridge University Press:  24 January 2017

V. L. CHERNYSHEV  and
A. A. TOLCHENNIKOV
V. L. CHERNYSHEV
Affiliation:
National Research University Higher School of Economics (HSE), Myasnitskaya Street, 20, Moscow, 101000, Russia email vchern@mech.math.msu.su
A. A. TOLCHENNIKOV
Affiliation:
A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Pr. Vernadskogo, 101-1, Moscow, 119526, Russia
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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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1697 - 1708
Copyright
© Cambridge University Press, 2017 

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