On some properties of the shift on an infinite-dimensional torus. (English. Russian original) Zbl 1527.37006
Differ. Equ. 59, No. 7, 867-879 (2023); translation from Differ. Uravn. 59, No. 7, 867-880 (2023).
Summary: We study the question, classical in the theory of dynamical systems, about the minimality of the shift on an infinite-dimensional torus; more precisely, the problem of finding sufficient conditions guaranteeing the absence of the minimality property is solved.
MSC:
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 58B25 | Group structures and generalizations on infinite-dimensional manifolds |
References:
| [1] | Smale, S., Differentiable dynamical systems, Usp. Mat. Nauk, 25, 1-151, 113-185 (1970) · Zbl 0205.54201 |
| [2] | Anosov, D. V.; Solodov, V. V., Hyperbolic sets, Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravl., 66, 12-99 (1991) |
| [3] | Anosov, D. V., Geodesic flows on closed Riemannian manifolds of negative curvature, Proc. Steklov Inst. Math., 90, 1-235 (1967) · Zbl 0176.19101 |
| [4] | Katok, A.; Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems (1995), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0878.58020 · doi:10.1017/CBO9780511809187 |
| [5] | Katok, A. B.; Khasselblat, B., A First Course in Dynamics: With a Panorama of Recent Developments (2003), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 1027.37001 |
| [6] | Pilyugin, S. Yu., Prostranstva dinamicheskikh sistem (Spaces of Dynamical Systems) (2008), Moscow-Izhevsk: Regulyarn. Khaoticheskaya Din., Moscow-Izhevsk |
| [7] | Grines, V. Z.; Pochinka, O. V., Vvedenie v topologicheskuyu klassifikatsiyu kaskadov na mnogoobraziyakh razmernosti dva ili tri (Introduction to the Topological Classification of Cascades on Manifolds of Dimension Two or Three) (2011), Moscow-Izhevsk: Regulyarn. Khaoticheskaya Din., Moscow-Izhevsk |
| [8] | Grines, V.; Zhuzhoma, E., Surface Laminations and Chaotic Dynamical Systems (2021), Moscow-Izhevsk: Izd. IKI, Moscow-Izhevsk · Zbl 1222.37040 |
| [9] | Palis, J.; de Melo, W., Geometric Theory of Dynamical Systems. An Introduction (1982), New York-Heidelberg-Berlin: Springer-Verlag, New York-Heidelberg-Berlin · Zbl 0491.58001 · doi:10.1007/978-1-4612-5703-5 |
| [10] | Pesin, Y. B., Lectures on Partial Hyperbolicity and Stable Ergodicity (2004), Zürich: Eur. Math. Soc., Zürich · Zbl 1098.37024 · doi:10.4171/003 |
| [11] | Robinson, C., Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Boca Raton: CRC Press, 1999 · Zbl 0914.58021 |
| [12] | Palis, J.; Takens, F., Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (1993), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0790.58014 |
| [13] | Ruelle, D., Large volume limit of the distribution of characteristic exponents in turbulence, Commun. Math. Phys., 87, 287-302 (1982) · Zbl 0546.76083 · doi:10.1007/BF01218566 |
| [14] | Mãné, R., Ergodic Theory and Differentiable Dynamics, Berlin-Heidelberg: Springer, 1987. · Zbl 0616.28007 |
| [15] | Thieullen, P., Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems, J. Dyn. Differ. Equat., 4, 1, 127-159 (1992) · Zbl 0744.34047 · doi:10.1007/BF01048158 |
| [16] | Hastings, H.M., On expansive homeomorphisms of the infinite torus, in The Structure of Attractors in Dynamical Systems. Lect. Notes Math., Berlin-Heidelberg-New York: Springer, 1978, pp. 142-149. · Zbl 0389.54022 |
| [17] | Mãné, R., Expansive homeomorphisms and topological dimension, Trans. Am. Math. Soc., 1979, vol. 252, pp. 313-319. · Zbl 0362.54036 |
| [18] | Glyzin, S. D.; Kolesov, A. Yu.; Rozov, N. Kh., Expansive endomorphisms on the infinite-dimensional torus, Funct. Anal. Appl., 54, 4, 241-256 (2020) · Zbl 1473.37024 · doi:10.1134/S0016266320040024 |
| [19] | Glyzin, S. D.; Kolesov, A. Yu.; Rozov, N. Kh., Solenoidal attractors of diffeomorphisms of annular sets, Russ. Math. Surv., 75, 2, 197-252 (2020) · Zbl 1448.37023 · doi:10.1070/RM9922 |
| [20] | Glyzin, S. D.; Kolesov, A. Yu.; Rozov, N. Kh., On a class of Anosov diffeomorphisms on the infinite-dimensional torus, Izv. Math., 85, 2, 177-227 (2021) · Zbl 1469.37021 · doi:10.1070/IM9002 |
| [21] | Glyzin, S. D.; Kolesov, A. Yu., A hyperbolicity criterion for a class of diffeomorphisms of an infinite-dimensional torus, Sb. Math., 213, 2, 173-215 (2022) · Zbl 1510.37048 · doi:10.1070/SM9535 |
| [22] | Glyzin, SD.; Kolesov, A. Yu., Elements of hyperbolic theory on an infinite-dimensional torus, Russ. Math. Surv., 77, 3, 379-443 (2022) · Zbl 1526.37033 · doi:10.1070/RM10058 |
| [23] | Jessen, B., The theory of integration in a space of an infinite number of dimensions, Acta Math., 63, 249-323 (1934) · Zbl 0010.20004 · doi:10.1007/BF02547355 |
| [24] | Platonov, S. S., Certain approximation problems for functions on the infinite-dimensional torus: Analogs of the Jackson theorem, St. Petersb. Math. J., 26, 6, 933-947 (2015) · Zbl 1326.41020 · doi:10.1090/spmj/1368 |
| [25] | Kosz, D., On differentiation of integrals in the infinite-dimensional torus, Studia Math., 258, 103-119 (2021) · Zbl 1466.43004 · doi:10.4064/sm191001-10-2 |
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