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Symbolic images and invariant measures of dynamical systems

Published online by Cambridge University Press:  17 July 2009

GEORGE OSIPENKO
GEORGE OSIPENKO*
Affiliation:
Department of Mathematics, St. Petersburg State Polytechnical University, St. Petersburg, Russia (email: [email protected])
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Abstract

Let f be a homeomorphism of a compact manifold M. The Krylov–Bogoloubov theorem guarantees the existence of a measure that is invariant with respect to f. The set of all invariant measures ℳ(f) is convex and compact in the weak topology. The goal of this paper is to construct the set ℳ(f). To obtain an approximation of ℳ(f), we use the symbolic image with respect to a partition C={M(1),M(2),…,M(n)} of M. A symbolic image G is a directed graph such that a vertex i corresponds to the cell M(i) and an edge i→j exists if and only if f(M(i))∩M(j)≠0̸. This approach lets us apply the coding of orbits and symbolic dynamics to arbitrary dynamical systems. A flow on the symbolic image is a probability distribution on the edges which satisfies Kirchhoff’s law at each vertex, i.e. the incoming flow equals the outgoing one. Such a distribution is an approximation to some invariant measure. The set of flows on the symbolic image G forms a convex polyhedron ℳ(G) which is an approximation to the set of invariant measures ℳ(f). By considering a sequence of subdivisions of the partitions, one gets sequence of symbolic images Gk and corresponding approximations ℳ(Gk) which tend to ℳ(f) as the diameter of the cells goes to zero. If the flows mk on each Gk are chosen in a special manner, then the sequence {mk} converges to some invariant measure. Every invariant measure can be obtained by this method. Applications and numerical examples are given.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 1217 - 1237
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Blank, M., Keller, G. and Liverani, C.. Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15 (2002), 1905–1973.CrossRefGoogle Scholar
[2]Bregman, L. M.. Relaxation method for determination of the common point of the convex sets and its application to problems of convex programming. J. Comput. Math. Math. Phys. 7(3) (1967), 630–631.Google Scholar
[3]Dellnitz, M. and Junge, O.. On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36(2) (1999), 491–515.CrossRefGoogle Scholar
[4]Ding, J. and Zhou, A.. Finite approximation of Frobenius–Perron operators, a solution of Ulam’s conjecture to multi-dimensional transformation. Phys. D 92(1–2) (1996), 61–66.CrossRefGoogle Scholar
[5]Froyland, G.. Finite approximation of Sinai–Bowen–Ruelle measures for Anosov systems in two dimensions. Random and Computational Dynamics 3(4) (1995), 251–263.Google Scholar
[6]Froyland, G.. Approximating physical invariant measures of mixing dynamical systems in higher dimensions. Nonlinear Anal. 32(7) (1998), 831–860.CrossRefGoogle Scholar
[7]Froyland, G.. Using Ulam’s method to calculate entropy and other dynamical invariants. Nonlinearity 12 (1999), 79–101.CrossRefGoogle Scholar
[8]Ikeda, K.. Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system. Opt. Comm. 30 (1979), 257–261.CrossRefGoogle Scholar
[9]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[10]Kryloff, N. and Bogollouboff, N.. La theorie generale de la mesure dans son application a l’etude das systemes dynamiques de la mecanique non lineaire. Ann. of Math. 38(1) (1937), 65–113.CrossRefGoogle Scholar
[11]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[12]Osipenko, G.. On a symbolic image of dynamical system. Boundary Value Problems. Interuniv. Collect. Sci. Works, Perm State Technical University, Perm, 1983, 101–105.Google Scholar
[13]Osipenko, G.. Numerical explorations of the Ikeda mapping dynamics. Electron J. Differential Equations and Control Processes 2 (2004), 43–67.  http://www.neva.ru/journal.Google Scholar
[14]Osipenko, G.. Dynamical Systems, Graphs, and Algorithms (Lecture Notes in Mathematics, 1889). Springer, Berlin, 2007.Google Scholar
[15]Sheleikhovsky, G. V.. Composition of city plan as a transport problem. Manuscript, 1946.Google Scholar
[16]Tien-Yien, L.. Finite approximation for the Frobenius–Perron operator, a solution to Ulam’s conjecture. J. Approx. Theory 17 (1976), 177–186.Google Scholar
[17]Ulam, S.. Problems in Modern Mathematics. Interscience, New York, 1960.Google Scholar