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 Mat. Sb., 1993, Volume 184, Number 8, Pages 17–36 (Mi msb1003)

Construction of an analogue of Bowen–Ruelle–Sinai (measure for a multidimensional lattice of interacting hyperbolic mappings

V. A. Volevich

Abstract: In this paper a dynamical system is studied whose phase space is an infinite product of finite-dimensional manifolds parametrized by the nodes of a multidimensional lattice and whose dynamics consists of a composition of hyperbolic mappings acting independently on each manifold and an interaction which introduces some dependence on adjacent variables. The interaction is assumed to be smooth and one-to-one. For such a dynamical system an invariant measure is constructed, and the system is shown to possess strong mixing properties, both in time and in space relative to this measure; i.e., the phenomenon of spatio-temporal chaos is observed. The idea of the proof is to construct a symbolic dynamics that makes it possible to apply results from the theory of Gibbs random fields.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1994, 79:2, 347–363

Bibliographic databases:

UDC: 517
MSC: Primary 28D05, 58F11, 58F15; Secondary 60G60, 58F13, 58F10

Citation: V. A. Volevich, “Construction of an analogue of Bowen–Ruelle–Sinai (measure for a multidimensional lattice of interacting hyperbolic mappings”, Mat. Sb., 184:8 (1993), 17–36; Russian Acad. Sci. Sb. Math., 79:2 (1994), 347–363

Citation in format AMSBIB
\Bibitem{Vol93} \by V.~A.~Volevich \paper Construction of an~analogue of Bowen--Ruelle--Sinai (measure for a~multidimensional lattice of interacting hyperbolic mappings \jour Mat. Sb. \yr 1993 \vol 184 \issue 8 \pages 17--36 \mathnet{http://mi.mathnet.ru/msb1003} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1239757} \zmath{https://zbmath.org/?q=an:0821.58027} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1994 \vol 79 \issue 2 \pages 347--363 \crossref{https://doi.org/10.1070/SM1994v079n02ABEH003504} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994PY27400007} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. J. Bricmont, A. Kupiainen, “High temperature expansions and dynamical systems”, Comm Math Phys, 178:3 (1996), 703
2. J. Bricmont, A. Kupiainen, “Infinite-dimensional SRB measures”, Physica D: Nonlinear Phenomena, 103:1-4 (1997), 18
3. Leonid A. Bunimovich, “Coupled map lattices: Some topological and ergodic properties”, Physica D: Nonlinear Phenomena, 103:1-4 (1997), 1
4. Baladi V., Esposti M., Isola S., Jarvenpaa E., Kupiainen A., “The Spectrum of Weakly Coupled Map Lattices”, J. Math. Pures Appl., 77:6 (1998), 539–584
5. Baladi V., Rugh H., “Floquet Spectrum of Weakly Coupled Map Lattices”, Commun. Math. Phys., 220:3 (2001), 561–582
6. Antônio M. Batista, Ricardo L. Viana, “Kolmogorov–Sinai entropy for locally coupled piecewise linear maps”, Physica A: Statistical Mechanics and its Applications, 308:1-4 (2002), 125
7. Jarvenpaa E., “Srb-Measures for Coupled Map Lattices”, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics, 671, eds. Chazottes J., Fernandez B., Springer-Verlag Berlin, 2005, 95–114
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