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 §(t)}t≥0 possesses a global
attractor A. We show that, for any generalized Banach limit
T→∞LIM and any distribution of initial
conditions m0, that there exists an invariant probability measure
m, whose support is contained in A, such that ∫Xϕ(x)dm(x)=T→∞LIMT1∫0T∫Xϕ(S(t)x)dm0(x)dt, for all
observables ϕ 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≥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.Comment: To appear in Communications in Mathematical Physic