More Ergodic Theorems

Towards a Koopman theory for dynamical systems on completely regular spaces

“…The first part readily follows from [28, Proposition 2.7]. The second part is a simple application of complete regularity of X analogous to the proof of [2, Lemma 4.18]. We omit the details.…”

“…Observe first that by Proposition 2.7 (iii) scriptTφ is strongly continuous on C b ( X ) if and only if double-struckR≥0false→Cfalse(Kfalse),1emtfalse↦Tφtffalse|K is continuous with respect to the norm topology on C( K ) for every compact subset K ⊆ X and every f ∈ C b ( X ). By [2, Lemma 4.16] this is equivalent to the continuity of the map double-struckR≥0×Kfalse→double-struckC,1emfalse(t,xfalse)false↦ffalse(φtfalse(xfalse)false) for every compact subset K ⊆ X and every f ∈ C b ( X ). Since X carries the initial topology with respect to the functions in C b ( X ), this simply means that the restricted mappings double-struckR≥0×Kfalse→X,1emfalse(t,xfalse)false↦φtfalse(xfalse) are continuous for every compact subset K ⊆ X .…”

“…continuous semiflows on topological spaces. In the case of dynamics on a compact space, this approach is systematically pursued in [2] for the time-discrete case and—to some extent—in [3] for the time-continuous one. At the heart of this ‘Koopman theory’ for topological dynamical systems lie on Banach algebras and Banach lattices, allowing a translation of properties of topological dynamical systems to the language of functional analysis.…”