Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

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 $\{S(t){\}}_{t\ge 0}$. Suppose that $\S (t){\}}_{t\ge 0}$ possesses a global attractor $\mathcal{A}$. We show that, for any generalized Banach limit $\underset{T\to \infty }{\mathrm{LIM}}$ and any distribution of initial conditions ${\mathfrak{m}}_0$, that there exists an invariant probability measure $\mathfrak{m}$, whose support is contained in $\mathcal{A}$, such that $${\int }_X\varphi (x)d\mathfrak{m}(x)=\underset{T\to \infty }{\mathrm{LIM}}\frac{1}{T}{\int }_0^T{\int }_X\varphi (S(t)x)d{\mathfrak{m}}_0(x)dt,$$ for all observables $\varphi$ living in a suitable function space of continuous mappings on $X$. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when $\{S(t){\}}_{t\ge 0}$ does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space $X$ to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.

To appear in Communications in Mathematical Physics

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