Geodesic
Construction of an analogue of Bowen–Ruelle–Sinai (measure for a multidimensional lattice of interacting hyperbolic mappings
Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 347-363 
V. A. Volevich. Construction of an analogue of Bowen–Ruelle–Sinai (measure for a multidimensional lattice of interacting hyperbolic mappings. Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 347-363. http://geodesic.mathdoc.fr/item/SM_1994_79_2_a6/
@article{SM_1994_79_2_a6,
     author = {V. A. Volevich},
     title = {Construction of an~analogue of {Bowen{\textendash}Ruelle{\textendash}Sinai} (measure for a~multidimensional lattice of interacting hyperbolic mappings},
     journal = {Sbornik. Mathematics},
     pages = {347--363},
     year = {1994},
     volume = {79},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_79_2_a6/}
}
TY  - JOUR
AU  - V. A. Volevich
TI  - Construction of an analogue of Bowen–Ruelle–Sinai (measure for a multidimensional lattice of interacting hyperbolic mappings
JO  - Sbornik. Mathematics
PY  - 1994
SP  - 347
EP  - 363
VL  - 79
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1994_79_2_a6/
LA  - en
ID  - SM_1994_79_2_a6
ER  - 
%0 Journal Article
%A V. A. Volevich
%T Construction of an analogue of Bowen–Ruelle–Sinai (measure for a multidimensional lattice of interacting hyperbolic mappings
%J Sbornik. Mathematics
%D 1994
%P 347-363
%V 79
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1994_79_2_a6/
%G en
%F SM_1994_79_2_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Pesin Ya. B., Sinai Ya. G., “Prostranstvenno-vremennoi khaos v tsepochkakh giperbolicheskikh otobrazhenii”, Advanced in Soviet mathematics, 3 (1991), 165–198 | MR | Zbl

[2] Volevich V. L., “Kinetics of Coupled map lattices”, Nonlinearity, 4 (1991), 37–48 | DOI | MR | Zbl

[3] Bunimovich L., Sinai Ya. G., “Space-time chaos in the Coupled map lattices”, Nonlinearity, 1 (1988), 491–516 | DOI | MR | Zbl

[4] Dobrushin R. L., Pecherski E. A., “Uniqueness conditions for finitely dependent Random Fields”, Random fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. Janos Bolyai, 27, North Holland, Amsterdam–Oxford–N.Y., 1981, 223–261 | MR

[5] Dobrushin R. L., “Zadanie sistemy sluchainykh velichin pri pomoschi uslovnykh raspredelenii”, Teoriya veroyat. i ee primen., XV:3 (1970), 469–497

[6] Sinai Ya. G., “Markovskie razbieniya i $Y$-diffeomorfizmy”, Funkts. analiz, 2:1 (1968), 64–89 | MR | Zbl

[7] Gundlash V. M., Rand D. A., Hyperbolicity Structural stability, Spatio-temporal Shadowing and Symbolic dynamics, Preprint, 1991

[8] Gundlash V. M., Rand D. A., Unique Gibbs state for Higher dimensional Symbolic system, Preprint, 1991

[9] Gundlash V. M., Rand D. A., Natural Spatio-temporal measures for Coupled circle map lattices, Preprint, 1991

[10] Dinamicheskie sistemy – 2, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 2, VINITI, M., 1985

[11] Bouen R., Metody simvolicheskoi dinamiki, Sb. statei, Mir, M., 1979

[12] Sinai Ya. G., “Gibbsovskie mery v ergodicheskoi teorii”, UMN, 27:4 (1972), 21–64 | MR | Zbl

[13] Kaneko K., “Towards Thermodynamics of Spatiotemporal Chaos”, Progress of Theoretical Physic Supplement, 99, 1989 | MR

[14] Dobrushin R. L., “Zadanie sistemy sluchainykh velichin pri pomoschi uslovnykh raspredelenii”, Teoriya veroyat. i ee primen., 15:3 (1970), 469–497 | MR | Zbl