A generic transformation is invertible

2025/12/22 by Tanja Eisner, Eisner, Tanja
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.2512.19893

openalex publication_date 2025/12/22 · openalex created_date 2025/12/25 · openalex updated_date 2026/07/28

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. We further show that invertible Koopman operators form a dense G_δ subset of all bi-stochastic operators for the weak operator topology, and the same holds for general Koopman operators.

Citations

• Generic extensions are recurrent for aperiodic probability automorphisms
• Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e.
• Self-joinings and generic extensions of ergodic systems
• Generic properties of l_p-contractions and similar operator topologies
• Generic measure preserving transformations and the closed groups they generate
• Does a typical \ℓ_p ,- ,space contraction have a non-trivial\n invariant subspace?
• Spectral theory of dynamical systems
• Relative weak mixing is generic

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