Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Graph Flows and Invariant Measures of Dynamical Systems

Author(s):

Georgii Sergeevich Osipenko

Professor of the Department of Applied Mathematics,
Branch of Lomonosov Moscow State University in Sevastopol

george.osipenko@mail.ru

Abstract:

We consider a discrete dynamical system generated by a homeomorphism f of a compact manifold. If {M (i)} is a finite covering of the manifold by closed cells, then there is a directed graph G with vertices corresponding to cells, and vertices i and j are connected by an edge i -> j if f(M(i)) intersects M(j). A periodic path on G generates a pseudotrajectory and a measure concentrated on it. Let a sequence of subdivisions with diameters converging to zero and a sequence of symbolic images be given. If a sequence of periodic paths is consistent then the corresponding sequence of periodic pseudotrajectories converges to a recurrent trajectory T, the sequence of measures converges to an ergodic measure and the closure of T is a minimal strictly ergodic set.

Keywords

• ergodicity
• flow on a graph
• of measures
• pseudotrajectory
• symbolic image
• weak convergence

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