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
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.
invariant measures, generalized functions, summation methods, small denominators, Hahn–Banach theorem.
|Russian Science Foundation
|This work is supported by the Russian Science Foundation under grant 14-50-00005.
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Izvestiya: Mathematics, 2016, 80:2, 342–358
MSC: 37C15, 37J15, 40G05, 46A22
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
\paper Invariant measures of smooth dynamical systems, generalized functions and summation methods
\jour Izv. RAN. Ser. Mat.
\jour Izv. Math.
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