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

This article is cited in 6 scientific papers (total in 6 papers)

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

Full text: PDF file (541 kB)
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English version:
Izvestiya: Mathematics, 2016, 80:2, 342–358

Bibliographic databases:

Document Type: Article
UDC: 519.21
MSC: 37C15, 37J15, 40G05, 46A22
Received: 10.11.2015

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

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    2. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. I. A. García, B. Hernández-Bermejo, “Inverse Jacobi multiplier as a link between conservative systems and Poisson structures”, J. Phys. A, 50:32 (2017), 325204, 17 pp.  crossref  mathscinet  zmath  isi  scopus
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    5. I. V. Volovich, V. Zh. Sakbaev, “On quantum dynamics on $C^*$-algebras”, Proc. Steklov Inst. Math., 301 (2018), 25–38  mathnet  crossref  crossref  isi  elib  elib
    6. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “An invariant measure and the probability of a fall in the problem of an inhomogeneous disk rolling on a plane”, Regul. Chaotic Dyn., 23:6 (2018), 665–684  mathnet  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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