A generic transformation is invertible

TL;DR

The paper investigates whether invertibility is a generic feature among measure-preserving transformations on a standard non-atomic probability space. It proves that the set of invertible transformations $\T_{inv}$ is dense and $G_\delta$ in the full transformation space $\u001aT$ under the strong (and hence weak) operator topology, implying that generic properties of invertible transformations extend to all measure-preserving transformations. The core methods include an operator-theoretic construction showing density of invertibles in the bi-stochastic operator closure via dyadic partitions, a separability-based Baire-category argument, and an alternative direct residuality proof. The results bridge the invertible and non-invertible regimes, enabling known generic phenomena for invertibles (e.g., entropy, rigidity, embedding into flows) to hold generically for all measure-preserving transformations, thereby broadening the applicability of generic ergodic properties.

Abstract

We show that, on a standard non-atomic probability space, invertible measure-preserving transformations form a dense $G_δ$ 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.

Theorems & Definitions (9)

• Theorem 1.1: Invertibility is generic
• Lemma 2.1
• proof
• Remark
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm:invgen']}