A generic transformation is invertible

TL;DR

The paper proves that invertible measure-preserving transformations are dense and generic within all measure-preserving transformations, and similarly for Koopman operators within bi-stochastic operators, under certain topologies.

Contribution

It establishes the density and genericity of invertible transformations and Koopman operators in their respective spaces under the strong and weak operator topologies.

Findings

Invertible measure-preserving transformations form a dense G_delta subset.

Invertible Koopman operators are dense in bi-stochastic operators.

Properties generic for invertible transformations are also generic for all measure-preserving transformations.

Abstract

We show that, on a standard non-atomic probability space, invertible measure-preserving transformations form a dense $G_\delta$ subset of the space of all measure-preserving transformations endowed with the strong (=weak) operator topology. This implies that all properties which are generic for invertible transformations are also generic for general ones. We further show that invertible Koopman operators form a dense $G_\delta$ subset of all bi-stochastic operators for the weak operator topology, and the same holds for general Koopman operators.