Invariant measures of smooth dynamical systems, generalized functions and summation methods

V. V. Kozlov

Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 342-358.

Résumé

We discuss conditions for the existence of invariant measures of smooth dynamical systems on compact manifolds. If there is an invariant measure with continuously differentiable density, then the divergence of the vector field along every solution tends to zero in the Cesàro sense as time increases unboundedly. Here the Cesàro convergence may be replaced, for example, by any Riesz summation method, which can be arbitrarily close to ordinary convergence (but does not coincide with it). We give an example of a system whose divergence tends to zero in the ordinary sense but none of its invariant measures is absolutely continuous with respect to the ‘standard’ Lebesgue measure (generated by some Riemannian metric) on the phase space. We give examples of analytic systems of differential equations on analytic phase spaces admitting invariant measures of any prescribed smoothness (including a measure with integrable density), but having no invariant measures with positive continuous densities. We give a new proof of the classical Bogolyubov–Krylov theorem using generalized functions and the Hahn–Banach theorem. The properties of signed invariant measures are also discussed.

Détail

Keywords: invariant measures, generalized functions, summation methods, small denominators, Hahn–Banach theorem.

Bibliographie

[1] V. V. Nemytskii, V. V. Stepanov, Qualitative theory of differential equations, Princeton Math. Ser., 22, Princeton Univ. Press, Princeton, NJ, 1960, viii+523 pp. | MR | MR | Zbl | Zbl

[2] P. R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, 3, Math. Soc. Japan, Tokyo, 1956, vii+99 pp. | MR | Zbl | Zbl

[3] A. N. Kolmogorov, “On dynamical systems with an integral invariant on a torus”, Selected works, vol. I: Mathematics and mechanics, Kluwer, Dordrecht, 1991, 344–348 | MR | Zbl | Zbl

[4] D. V. Anosov, “On an additive functional homology equation connected with an ergodic rotation of the circle”, Math. USSR-Izv., 7:6 (1973), 1257–1271 | DOI | MR | Zbl

[5] A. Ya. Gordon, “Sufficient condition for unsolvability of the additive functional homological equation connected with the ergodic rotation of a circle”, Funct. Anal. Appl., 9:4 (1975), 334–336 | DOI | MR | Zbl

[6] V. V. Kozlov, “Integrability and non-integrability in Hamiltonian mechanics”, Russian Math. Surveys, 38:1 (1983), 1–76 | DOI | MR | Zbl

[7] N. G. Moshchevitin, “Existence and smoothness of the integral of a Hamiltonian system of a certain form”, Math. Notes, 49:5 (1991), 498–501 | DOI | MR | Zbl

[8] C. Grebogi, E. Ott, S. Pelikan, J. A. Yorke, “Strange attractors that are not chaotic”, Phys. D, 13:1-2 (1984), 261–268 | DOI | MR | Zbl

[9] V. V. Kozlov, “On integrals of quasiperiodic functions”, Mosc. Univ. Mech. Bull., 33:1-2 (1978), 31–38 | MR | Zbl

[10] U. Feudel, S. Kuznetsov, A. Pikovsky, Strange nonchaotic attractors. Dynamics between order and chaos in quasiperiodically forced systems, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006, xii+213 pp. | MR | Zbl

[11] J. C. Sprott, Elegant chaos. Algebraically simple chaotic flows, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010, xv+285 pp. | MR | Zbl

[12] D. Topaj, A. Pikovsky, “Reversibility vs. synchronization in oscillator lattices”, Phys. D, 170:2 (2002), 118–130 | DOI | MR | Zbl

[13] V. V. Kozlov, “On the existence of an integral invariant of a smooth dynamic system”, J. Appl. Math. Mech., 51:4 (1987), 420–426 | DOI | MR | Zbl

[14] Ya. G. Sinai, Introduction to ergodic theory, Math. Notes, 18, Princeton Univ. Press, Princeton, NJ, 1976, 144 pp. | MR | Zbl

[15] O. Veblen, Invariants of quadratic differential forms, Cambridge Tracts in Mathematics and Mathematical Physics, 24, Cambridge Univ. Press, Cambridge, 1927, viii+102 pp. | Zbl

[16] P. K. Rashevskii, Geometricheskaya teoriya uravnenii s chastnymi proizvodnymi, Gostekhizdat, M.–L., 1947, 354 pp. | MR | Zbl

[17] V. Arnol'd, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir, M., 1980, 324 pp. | MR | MR | Zbl | Zbl

[18] G. H. Hardy, Divergent series, Oxford, Clarendon Press, 1949, xvi+396 pp. | MR | MR | Zbl

[19] V. V. Kozlov, “Uniform distribution on a torus”, Mosc. Univ. Math. Bull., 59:2 (2004), 23–31 | MR | Zbl

[20] A. Yu. Obolenskii, Lektsii po kachestvennoi teorii differentsialnykh uravnenii, Regulyarnaya i khaoticheskaya dinamika, In-t kompyuternykh issledovanii, M.–Izhevsk, 2006, 320 pp.

[21] N. Bourbaki, Éléments de mathématique. Première partie: Les structures fondamentales de l'analyse. Livre V: Espaces vectoriels topologiques, Ch. I–V, v. XV, XVIII, XIX, Actualités Sci. Indust., 1189, 1229, 1230, Hermann, Paris, 1953, 1955, 1955, 123 pp., 190 pp., ii+391 pp. | MR | Zbl

[22] O. Zubelevich, “Several notes on existence theorem of Peano”, Funkcial. Ekvac., 55:1 (2012), 89–97 | DOI | MR | Zbl

[23] V. V. Kozlov, “K teorii integrirovaniya uravnenii negolonomnoi mekhaniki”, Uspekhi mekhaniki, 8:3 (1985), 85–107 | MR

[24] A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Hamiltonicity and integrability of the Suslov problem”, Regul. Chaotic Dyn., 16:1-2 (2011), 104–116 | DOI | MR | Zbl