Invertible measure-preserving transformations form a dense Gδ subset of all measure-preserving transformations on standard non-atomic probability spaces under the strong operator topology.
V Ryzhikov,Generic properties of ergodic automorphisms, preprint, see https://arxiv.org/abs/2407.18236
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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.
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A generic transformation is invertible
Invertible measure-preserving transformations form a dense Gδ subset of all measure-preserving transformations on standard non-atomic probability spaces under the strong operator topology.
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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.