The weak limit semigroup of an operator T is the collection of all weak limit points of its powers, and the paper shows that this collection contains large subsets in generic cases for Koopman, contraction, and positive operators.
Ageev,On the genericity of some non-asymptotic dynamical properties, Uspekhi Mat
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Weak limit semigroup in operator theory and ergodic theory
The weak limit semigroup of an operator T is the collection of all weak limit points of its powers, and the paper shows that this collection contains large subsets in generic cases for Koopman, contraction, and positive operators.