Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

Citation:

Chekroun, Mickaël D., and Nathan E. Glatt-Holtz. 2012. “Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications.” Communications in Mathematical Physics 316 (3): 723–761.

Abstract:

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space X which is acted on by any continuous semigroup \S(t)\ t ≥ 0. Suppose that \S(t)\ t ≥ 0 possesses a global attractor \$\$\\backslashmathcal\A\\\$\$ . We show that, for any generalized Banach limit LIM T → ∞ and any probability distribution of initial conditions \$\$\\backslashmathfrak\m\\_0\\$\$ , that there exists an invariant probability measure \$\$\\backslashmathfrak\m\\\$\$ , whose support is contained in \$\$\\backslashmathcal\A\\\$\$ , such that \$\$\backslashint\_\X\ \backslashvarphi(x) \\backslashrm d\\backslashmathfrak\m\(x) = \backslashunderset\t \backslashrightarrow \backslashinfty\\\backslashrm LIM\\backslashfrac\1\\T\ \backslashint\_0^T \backslashint\_X \backslashvarphi(S(t) x) \\backslashrm d\\backslashmathfrak\m\\_0(x) \\backslashrm d\t,\$\$ for all observables $\phi$ living in a suitable function space of continuous mappings on X.

Publisher's Version

Last updated on 04/19/2020