pith. sign in

arxiv: 2512.19893 · v5 · submitted 2025-12-22 · 🧮 math.DS · math.FA

A generic transformation is invertible

Tanja Eisner This is my paper

Pith reviewed 2026-05-16 20:01 UTC · model grok-4.3

classification 🧮 math.DS math.FA
keywords measure-preserving transformationsinvertible transformationsdense Gδ setstrong operator topologyKoopman operatorsbi-stochastic operatorsgeneric properties
0
0 comments X

The pith

On standard non-atomic probability spaces, invertible measure-preserving transformations form a dense Gδ subset of all such transformations in the strong operator topology.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that invertible measure-preserving transformations are a dense Gδ set in the space of all measure-preserving transformations equipped with the strong operator topology, provided the probability space is standard and non-atomic. This implies that any property generic for invertible transformations will also be generic for general transformations. The same holds for invertible Koopman operators among bi-stochastic operators in the weak operator topology. Readers interested in dynamical systems would care because this unifies the study of generic properties across invertible and non-invertible cases without separate proofs.

Core claim

We show that, on a standard non-atomic probability space, invertible measure-preserving transformations form a dense Gδ 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 Gδ subset of all bi-stochastic operators for the weak operator topology, and the same holds for general Koopman operators.

What carries the argument

The strong operator topology on the space of measure-preserving transformations (coinciding with the weak operator topology), under which the invertible ones form a dense Gδ set.

If this is right

  • Any property that holds for a comeager set of invertible maps, such as ergodicity or weak mixing, also holds for a comeager set of all maps.
  • Results established only for invertible transformations transfer directly to the general case for generic statements.
  • The same density and genericity conclusion applies to invertible Koopman operators inside the space of bi-stochastic operators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Non-invertible transformations can be approximated arbitrarily closely by invertible ones in the strong topology, which may simplify perturbation arguments in examples.
  • The unification allows generic statements in ergodic theory to be stated once for the invertible setting and applied broadly.
  • Numerical approximations or simulations of generic dynamics can assume invertibility without loss of generality due to the density.

Load-bearing premise

The underlying probability space is standard and non-atomic.

What would settle it

An explicit construction of a standard non-atomic probability space in which the invertible transformations fail to be dense under the strong operator topology would disprove the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that, on a standard non-atomic probability space, the invertible measure-preserving transformations form a dense Gδ subset of all measure-preserving transformations when the space is equipped with the strong operator topology (which is shown to coincide with the weak operator topology). It further establishes that invertible Koopman operators form a dense Gδ subset of all bi-stochastic operators under the weak operator topology. The result implies that properties generic among invertible transformations are also generic among general ones.

Significance. If the result holds, it is significant for ergodic theory because it shows that the invertible case is comeager in the larger space of all measure-preserving transformations, allowing generic properties established for invertible maps to transfer directly to the non-invertible setting. The argument relies on standard Rokhlin-tower approximations to establish density while controlling distances in the strong operator topology, together with a countable-intersection representation of invertibility to obtain the Gδ property. The explicit identification of the strong and weak topologies on the set of MPTs is a useful technical observation that simplifies the topology.

minor comments (2)
  1. [Introduction] The parenthetical claim that the strong operator topology equals the weak operator topology on MPTs is stated without a self-contained reference or short argument; adding a one-sentence justification (e.g., via the multiplicative property of Koopman operators) would improve readability.
  2. [Main Theorem] In the statement of the main theorem, the precise definition of the strong operator topology (via L² convergence on a dense set of functions) should be recalled explicitly rather than assumed from prior literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the recognition of the result's significance for ergodic theory, and the recommendation of minor revision. We appreciate the note that the identification of the strong and weak topologies is a useful technical observation.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central claim is proved via standard Rokhlin-tower approximations to establish density of invertible MPTs and by expressing invertibility as a countable intersection of open sets in the strong operator topology to obtain the Gδ property. These steps rely on the non-atomic assumption for flexible approximations and on the multiplicative structure of Koopman operators; neither reduces to a fitted parameter, self-definition, or load-bearing self-citation. The equality of strong and weak topologies on MPTs follows directly from the definition of the topologies and the preservation of integrals under measure-preserving maps. No equation or step in the derivation chain is equivalent to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of a non-atomic probability space and the identification of strong and weak operator topologies for these operators; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The probability space is standard and non-atomic
    Explicitly required for the main density statement.
  • standard math Strong operator topology coincides with weak operator topology on the relevant operator space
    Stated directly in the abstract as strong (=weak).

pith-pipeline@v0.9.0 · 5360 in / 1080 out tokens · 28542 ms · 2026-05-16T20:01:10.483434+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Weak limit semigroup in operator theory and ergodic theory

    math.FA 2026-03 unverdicted novelty 5.0

    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 positi...

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [1]

    O. N. Ageev,On the genericity of some non-asymptotic dynamical properties, Uspekhi Mat. Nauk58(2003), 177–178; translation in Russian Math. Surveys58(2003), 173–174

    work page 2003

  2. [2]

    R. V. Chacon,Weakly mixing transformations which are not strongly mixing, Proc. Am. Math. Soc.22(1969), 559–562

    work page 1969

  3. [3]

    J. R. Choksi, M. G. Nadkarni,Baire category in spaces of measures, unitary operators, and transformations, Invariant Subspaces and Allied Topics, pp. 147–163. Narosa Publishing House (1990)

    work page 1990

  4. [4]

    de la Rue, J

    T. de la Rue, J. de Sam Lazaro,Une transformation générique peut être insérée dans un flot, Ann. Inst. H. Poincaré Probab. Stat.39(2003), 121–134

    work page 2003

  5. [5]

    del Junco, M

    A. del Junco, M. Lemańczyk,Generic spectral properties of measure-preserving maps and applications, Proc. Amer. Math. Soc.115(1992), 725–736

    work page 1992

  6. [6]

    Eisner,A “typical” contraction is unitary, Enseign

    T. Eisner,A “typical” contraction is unitary, Enseign. Math.56(2010), 403–410

    work page 2010

  7. [7]

    Eisner, B

    T. Eisner, B. Farkas, A Journey Through Ergodic Theorems, Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser Verlag, 2025

    work page 2025

  8. [8]

    Glasner, B

    E. Glasner, B. Weiss,Relative weak mixing is generic, Sci. China Math.62(2019), 69–72

    work page 2019

  9. [9]

    Grivaux, É

    S. Grivaux, É. Matheron, Q. Menet,Does a typicallp-space contraction have a non-trivial invariant subspace?, Trans. Amer. Math. Soc.374(2021), 7359–7410

    work page 2021

  10. [10]

    Grivaux, É

    S. Grivaux, É. Matheron, Q. Menet,Generic properties oflp-contractions and similar oper- ator topologies. Preprint, 2022, available at https://arxiv.org/abs/2207.07938

    work page arXiv 2022

  11. [11]

    P. R. Halmos,In general a measure preserving transformation is mixing, Ann. Math.45 (1944), 786–792

    work page 1944

  12. [12]

    Halmos, Lectures on Ergodic Theory, Chelsea Publishing Company, 1956

    P. Halmos, Lectures on Ergodic Theory, Chelsea Publishing Company, 1956

    work page 1956

  13. [13]

    Katok,Approximation and Generity in Abstract Ergodic Theory, Notes 1985

    A. Katok,Approximation and Generity in Abstract Ergodic Theory, Notes 1985

    work page 1985

  14. [14]

    Kechris, Classical Descriptive Set Theory

    A.S. Kechris, Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156, Springer, 1995

    work page 1995

  15. [15]

    J. L. King,The generic transformation has roots of all orders, Colloq. Math.84/85(2000), 521–547

    work page 2000

  16. [16]

    D. K. Kozyreva, V. V Ryzhikov,Generic extensions are recurrent for aperiodic probability automorphisms, preprint, see https://arxiv.org/abs/2510.16006

    work page arXiv

  17. [17]

    M. G. Nadkarni, Spectral Theory of Dynamical Systems. Birkhäuser Advanced Texts, Basler Lehrbücher, 1991

    work page 1991

  18. [18]

    V. A. Rohlin,A “general” measure-preserving transformation is not mixing, Doklady Akad. Nauk SSSR60(1948), 349–351

    work page 1948

  19. [19]

    V. A. Rohlin,Entropy of metric automorphism, Doklady Akad. Nauk SSSR124(1959), 980–983

    work page 1959

  20. [20]

    Ryzhikov,Intertwinings of tensor products, and the stochastic centralizer of dynamical systems, Sb

    V. Ryzhikov,Intertwinings of tensor products, and the stochastic centralizer of dynamical systems, Sb. Math.,188(1997), 237–263

    work page 1997

  21. [21]

    Ryzhikov,Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions, Sb

    V. Ryzhikov,Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions, Sb. Math.198(2007), 733–754

    work page 2007

  22. [22]

    V Ryzhikov,Generic extensions of ergodic systems, Mat

    V. V Ryzhikov,Generic extensions of ergodic systems, Mat. Sb.214(2023), 98–115; trans- lation in Sb. Math.214(2023), 1442–1457

    work page 2023

  23. [23]

    V Ryzhikov,Self-joinings and generic extensions of ergodic systems, Translation of Funk- tsional

    V. V Ryzhikov,Self-joinings and generic extensions of ergodic systems, Translation of Funk- tsional. Anal. i Prilozhen.57(2023), 74–88, Funct. Anal. Appl.57(2023), 236–247

    work page 2023

  24. [24]

    V Ryzhikov,Generic properties of ergodic automorphisms, preprint, see https://arxiv.org/abs/2407.18236

    V. V Ryzhikov,Generic properties of ergodic automorphisms, preprint, see https://arxiv.org/abs/2407.18236

    work page arXiv

  25. [25]

    Schnurr,Generic properties of extensions, Ergodic Theory Dynam

    M. Schnurr,Generic properties of extensions, Ergodic Theory Dynam. Systems39(2019), 3144–3168

    work page 2019

  26. [26]

    Solecki,Closed subgroups generated by generic measure automorphisms, Ergodic Theory Dynam

    S. Solecki,Closed subgroups generated by generic measure automorphisms, Ergodic Theory Dynam. Systems34(2014), 1011–1017

    work page 2014

  27. [27]

    Solecki,Generic measure preserving transformations and the closed groups they generate, Invent

    S. Solecki,Generic measure preserving transformations and the closed groups they generate, Invent. Math.231(2023), 805–850

    work page 2023

  28. [28]

    A. M. Stepin,Spectral properties of generic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 801–834, 879

    work page 1986

  29. [29]

    A. M. Stepin, A. M. Eremenko,Nonuniqueness of an inclusion in a flow and the vastness of a centralizer for a generic measure-preserving transformation, Mat. Sb. 195(2004), 95–108; translation in Sb. Math. 195(2004), 1795–1808. 6 TANJA EISNER

    work page 2004

  30. [30]

    A. M. Vershik,Multivalued measure-preserving mappings (polymorphisms) and Markov pro- cesses, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 72 (1977), 26–62; English transl. in J. Soviet Math. 23:4 (1983). Institute of Mathematics, University of Leipzig, P.O. Box 100 920, 04009 Leipzig, Germany Email address:eisner@math.uni-leipzig.de

    work page 1977