(Differencialnie Uravnenia i Protsesy Upravlenia)

About
History
Editorial Page
Addresses
Scope
Editorial Staff
Submission Review
For Authors
Publication Ethics
Issues
Русская версия

**Georgii Sergeevich Osipenko**

Professor of the Department of Applied Mathematics,

Branch of Lomonosov Moscow State University in Sevastopol

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.

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

- M. Shub. Stabilite globale de systems denamiques. // Asterisque, 1978, v. 56, 1-21
- G. D. Birkhoff. Proof of recurrence theorem for strongly transitive systems. Proof of the ergodic theorem. // Proc. Nat. Acad. Sci. v. 17, 1931
- G. S. Osipenko, On the symbolic image of a dynamical system. // Boundary Value Problems. Interuniv. Collect. Sci. Works, Perm State Technical University, Perm, 1983, pp. 101-105 [in Russian]
- George Osipenko. Dynamical systems, Graphs, and Algorithms. Lectures Notes in Mathematics, v. 1889, Springer, Berlin, 2007
- V. M. Alekseev, Symbolic Dynamics, 11
^{th}Mathematical School, Kiev, 1976 (in Russian) - Lind Douglas, Marcus Brian. An introduction to symbolic dynamics and coding. Cambridge University Press, 1995
- C. Robinson. Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 1995
- C. S. Hsu. Cell-to-Cell Mapping, Springer-Verlag, N. Y. 1987
- V. Prasolov, Elementi combinatornoii and differentialnoi topologii, M., izd. MCNMO , 2004 (in Russian)
- G. S. Osipenko, Encodings of trajectories and invariant measures. // Sbornik: Mathematics 211:7 (2020), 1041-1064
- George Osipenko. Symbolic images and invariant measures of dynamical systems. // Ergodic Theory and Dynamical Systems. v. 30, 2010, 1217 - 1237
- A. Katok, B. Hasselblat, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995
- V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Dynamical Systems, Princeton University Press, 1960. Russian original 1949
- G. S. Osipenko, Mean convergence of periodic pseudotrajectories and invariant measures of dynamical systems, // Mathematical Notes, 2020, Vol. 108, No. 6, pp. 854-866. Pleiades Publishing, Ltd., 2020. Russian Text 2020, published in Matematicheskie Zametki, 2020, Vol. 108, No. 6, pp. 882-898