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Sbornik: Mathematics, 2020, Volume 211, Issue 7, Pages 1041–1064
DOI: https://doi.org/10.1070/SM9273
(Mi sm9273)
 

Encodings of trajectories and invariant measures

G. S. Osipenko

Sevastopol Branch of Lomonosov Moscow State University
References:
Abstract: We consider a discrete dynamical system on a compact manifold $M$ generated by a homeomorphism $f$. Let $C=\{M(i)\}$ be a finite covering of $M$ by closed cells. The symbolic image of a dynamical system is a directed graph $G$ with vertices corresponding to cells in which vertices $i$ and $j$ are joined by an arc $i\to j$ if the image $f(M(i))$ intersects $M(j)$. We show that the set of paths of the symbolic image converges to the set of trajectories of the system in the Tychonoff topology as the diameter of the covering tends to zero. For a cycle on $G$ going through different vertices, a simple flow is by definition a uniform distribution on arcs of this cycle. We show that simple flows converge to ergodic measures in the weak topology as the diameter of the covering tends to zero.
Bibliography: 28 titles.
Keywords: pseudotrajectory, recurrent trajectory, chain recurrent set, ergodic measure, symbolic image, flow on a graph.
Funding agency Grant number
Russian Foundation for Basic Research 19-01-00388-а
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 19-01-00388-a).
Received: 27.04.2019 and 12.02.2020
Bibliographic databases:
Document Type: Article
UDC: 517.938
MSC: 37C50
Language: English
Original paper language: Russian
Citation: G. S. Osipenko, “Encodings of trajectories and invariant measures”, Sb. Math., 211:7 (2020), 1041–1064
Citation in format AMSBIB
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\paper Encodings of trajectories and invariant measures
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\yr 2020
\vol 211
\issue 7
\pages 1041--1064
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  • https://doi.org/10.1070/SM9273
  • https://www.mathnet.ru/eng/sm/v211/i7/p151
  • Citing articles in Google Scholar: Russian citations, English citations
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