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 Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 2, Pages 63–80 (Mi izv8469)

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

V. V. Kozlov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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.

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

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.4213/im8469

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English version:
Izvestiya: Mathematics, 2016, 80:2, 342–358

Bibliographic databases:

UDC: 519.21
MSC: 37C15, 37J15, 40G05, 46A22

Citation: V. V. Kozlov, “Invariant measures of smooth dynamical systems, generalized functions and summation methods”, Izv. RAN. Ser. Mat., 80:2 (2016), 63–80; Izv. Math., 80:2 (2016), 342–358

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8469
• https://doi.org/10.4213/im8469
• http://mi.mathnet.ru/eng/izv/v80/i2/p63

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