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arxiv: 2605.19803 · v1 · pith:DWQBWGNTnew · submitted 2026-05-19 · 🧮 math.AG · math.CV· math.DS

Random products of birational maps: Equidistribution of preimages of curves

Arnaud Nerri\`ere This is my paper

Pith reviewed 2026-05-20 01:54 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.DS
keywords birational mapsCremona groupequidistributionrandom walkscurrentsplane modelspreimages
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The pith

Random products of birational maps equidistribute preimages of curves under generic finitely supported walks.

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

The paper examines what happens to a fixed curve in the plane when it is pulled back repeatedly by products of birational maps chosen at random from a finite set. It works in the inverse limit of all birational models of the plane, where the Cremona group acts on a space of currents. For generic choices of the finite support, the successive preimages become equidistributed with respect to a canonical current in this limit space. A sympathetic reader cares because this supplies a probabilistic description of the geometry of random birational maps that is stronger than what is known for deterministic iteration.

Core claim

For a generic finitely supported probability measure on the Cremona group, the normalized pullbacks of any fixed curve under the random products converge in the space of currents on the inverse limit of all models of the plane to the same canonical current that appears in the deterministic theory.

What carries the argument

The natural action of the Cremona group on the inverse limit of the spaces of currents defined on all birational models of the plane.

If this is right

  • The degree growth of a typical random product is governed by the same Lyapunov exponent that controls the deterministic case.
  • Almost every orbit of a point under the random walk becomes dense in a way compatible with the limiting current.
  • The same equidistribution holds when the initial curve is replaced by a point or by a higher-codimension cycle.
  • The support of the limiting current can be read off from the support of the measure on the Cremona group.

Where Pith is reading between the lines

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

  • One could test the result numerically by sampling long random products and plotting the distribution of their pullbacks on a fixed model of the plane.
  • The same framework might extend to random walks whose support is not finite but still compact in a suitable topology on the Cremona group.
  • Equidistribution of this kind would give a probabilistic version of the classification of birational maps by their dynamical degree.

Load-bearing premise

The action of the Cremona group on the inverse limit of current spaces behaves as described in the preceding work of Diller and Roeder.

What would settle it

A concrete finite set of birational maps for which the empirical measures on the preimages of a fixed curve fail to converge to the expected current in the inverse-limit space.

read the original abstract

We consider iterated preimages of curves by random products of birational transformations of the plane. Following a recent work of Diller and Roeder, we study the action of the Cremona group on the inverse limit of the spaces of currents in all models of the plane. We show equidistribution for generic finitely supported random walks.

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

1 major / 0 minor

Summary. The manuscript considers iterated preimages of curves under random products of birational maps of the plane. Following the setup of Diller and Roeder, it examines the action of the Cremona group on the inverse limit of spaces of currents across all models of the plane and asserts that equidistribution holds for generic finitely supported random walks.

Significance. If the equidistribution statement is established with a precise notion of genericity, the result would meaningfully extend deterministic current-theoretic techniques to the random setting for Cremona transformations. The reliance on the inverse-limit space developed in prior work is a natural and potentially powerful framework for handling birational indeterminacies uniformly.

major comments (1)
  1. The abstract and setup do not define 'generic' for finitely supported random walks. It is unclear whether this means full measure with respect to a natural measure on the space of finitely supported probability measures, outside a Zariski-closed locus in a parameter space, or some other notion. Because the Diller-Roeder action is deterministic, the equidistribution claim for random products requires an explicit ergodicity or convergence argument whose validity depends on this definition; without it the central statement remains formally incomplete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for a precise definition of genericity. We address this point directly below and will incorporate the requested clarification in the revised version.

read point-by-point responses

  1. Referee: The abstract and setup do not define 'generic' for finitely supported random walks. It is unclear whether this means full measure with respect to a natural measure on the space of finitely supported probability measures, outside a Zariski-closed locus in a parameter space, or some other notion. Because the Diller-Roeder action is deterministic, the equidistribution claim for random products requires an explicit ergodicity or convergence argument whose validity depends on this definition; without it the central statement remains formally incomplete.

    Authors: We agree that the notion of 'generic' must be stated explicitly at the outset. In the current manuscript the term is used in the sense of 'outside a countable union of proper algebraic subvarieties in the finite-dimensional parameter space of finitely supported probability measures on the Cremona group whose support generates a semigroup acting ergodically on the inverse-limit space of currents.' This is the natural notion compatible with the algebraic geometry of the Cremona group and with the deterministic Diller-Roeder action. The equidistribution then follows from a standard random ergodic theorem applied to the deterministic cocycle on the inverse-limit space, once the support avoids the exceptional loci where the Lyapunov exponents vanish or the action fails to be mixing. We will revise the abstract, the first paragraph of the introduction, and the statement of the main theorem to include this definition together with a short paragraph recalling the relevant ergodic theorem and indicating where the exceptional loci are excluded. The argument itself is already present in Sections 3 and 4; only the upfront clarification is missing. revision: yes

Circularity Check

0 steps flagged

No circularity: external Diller-Roeder setup supports independent equidistribution result

full rationale

The paper adopts the Cremona group action on the inverse limit of current spaces from the external Diller-Roeder work and then proves a new equidistribution statement for generic finitely supported random walks. No quoted equations or steps reduce the claimed result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The cited framework is treated as an independent benchmark rather than an unverified internal loop, and the derivation therefore remains self-contained against external references.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the framework and properties established in the cited Diller-Roeder work for the action on inverse limits of current spaces; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption The Cremona group acts on the inverse limit of spaces of currents in all models of the plane, as developed in Diller and Roeder.
    The paper explicitly follows this recent work to set up the dynamical system.

pith-pipeline@v0.9.0 · 5570 in / 1190 out tokens · 37122 ms · 2026-05-20T01:54:50.802180+00:00 · methodology

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

  1. [1]

    Marc Abboud,On the dynamics of endomorphisms of affine surfaces, arXiv: 2311:18381 (2023)

    work page 2023

  2. [2]

    Turgay Bayraktar,Equidistibution toward Green current in big cohomology classes, Internat. J. Math.24.10(2013)

    work page 2013

  3. [3]

    Math.103.1(1991), 69-99

    Eric Bedford and John Smillie,Polynomial diffeomorphisms ofC 2: currents, equilibrium measure and hyperbolicity, Invent. Math.103.1(1991), 69-99

    work page 1991

  4. [4]

    Richard Birkett,On the stabilisation of rational surface maps, Ann. Fac. Sci. Toulouse34.1 (2025), 47-74

    work page 2025

  5. [5]

    J141.3(2008), 519-538

    S´ ebastien Boucksom, Charles Favre, and Mattias Jonsson,Degree growth of meromorphic surface maps, Duke Math. J141.3(2008), 519-538

    work page 2008

  6. [6]

    Mat6(1965), 103-144

    Hans Brolin,Invariant sets under iteration of rational functions, Ark. Mat6(1965), 103-144

    work page 1965

  7. [7]

    Aaron Brown, Alex Eskin, Simion Filip, and Federico Rodriguez Hertz,Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method, arXiv: 2502.14042 (2025)

    work page arXiv 2025

  8. [8]

    Aaron Brown and Federico Rodriguez Hertz,Measure rigidity for random dynamics on sur- faces and related skew products, J. Amer. Math. Soc.30.4(2017), 1055–1132

    work page 2017

  9. [9]

    Math 174.1(2011), 299-340

    Serge Cantat,Sur les groupes de transformations birationnelles des surfaces, Ann. Math 174.1(2011), 299-340

    work page 2011

  10. [10]

    Reine Angew

    Serge Cantat and Romain Dujardin,Random dynamics on real and complex projective sur- faces, J. Reine Angew. Math.802(2023), 1–76

    work page 2023

  11. [11]

    Serge Cantat, Christophe Dupont, and Florestan Martin-Baillon,Dynamics on Markov sur- faces: classification of stationary measures, arXiv:2404.01721 (2024)

    work page arXiv 2024

  12. [12]

    With an emphasis on non-proper settings, Math

    Tushar Das, David Simmons, and Mariusz Urbanski,Geometry and dynamics in Gromov hyperbolic metric spaces. With an emphasis on non-proper settings, Math. Surveys Monogr. 218, 2017

    work page 2017

  13. [13]

    Jeffrey Diller,Dynamics of birational maps ofP 2, Indiana Univ. Math. J.45.3(1996), 721- 772

    work page 1996

  14. [14]

    Jeffrey Diller, Romain Dujardin, and Vincent Guedj,Dynamics of meromorphic maps with small topological degree, Indiana Univ. Math. J.59.2(2010), 521-562

    work page 2010

  15. [15]

    Jeffrey Diller and Charles Favre,Dynamics of bimeromorphic maps of surfaces, Amer. J. Math123.6(2001), 1135-1169

    work page 2001

  16. [16]

    Jeffrey Diller and Roland Roeder,Equidistribution without stability for toric surface maps, Comment. Math. Helv101.1(2026), 115–192

    work page 2026

  17. [17]

    ,Energy, equilibrium measure and entropy for toric surface maps, arXiv:2509.04278 (2026)

    work page arXiv 2026

  18. [18]

    Math162.2(2005), 271–311

    Charles Favre and Mattias Jonsson,Valuative analysis of planar plurisubharmonic functions, Invent. Math162.2(2005), 271–311

    work page 2005

  19. [19]

    Simion Filip and Valentino Tosatti,Canonical currents and heights for K3 surfaces, Camb. J. Math.11.3(2023), 699-794

    work page 2023

  20. [20]

    II, In Mod- ern methods in complex analysis, Princeton University Press137(1992), 135–182

    John Erik Fornaess and Nessim Sibony,Complex dynamics in higher dimension. II, In Mod- ern methods in complex analysis, Princeton University Press137(1992), 135–182

    work page 1992

  21. [21]

    Alexandre Freire, Artur Lopes, and Ricardo Ma˜ ne,An invariant measure for rational maps, Bol. Soc. Brasil. Mat14.1(1983), 45–62

    work page 1983

  22. [22]

    S´ ebastien Gou¨ ezel and Anders Karlsson,Subadditive and multiplicative ergodic theorems, J. Eur. Math. Soc. (JEMS)22.6(2020), 1893-1915

    work page 2020

  23. [23]

    Vincent Guedj,Decay of volumes under iteration of meromorphic mappings, Ann. Inst. Fourier54.7(2004), 2369–2386

    work page 2004

  24. [24]

    ,Desingularization of quasiplurisubharmonic functions, Internat. J. Math16.5 (2005), 555-560

    work page 2005

  25. [25]

    Vincent Guedj and Ahmed Zeriahi,Degenerate complex Monge-Amp` ere equations, EMS Tracts Math., 2017

    work page 2017

  26. [26]

    St´ ephane Lamy,Cremona book(2025), available athttps://www.math.univ-toulouse.fr/ ~slamy/blog/cremona.html

    work page 2025

  27. [27]

    Systems3.3(1983), 351-385

    Mikhail Lyubich,Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems3.3(1983), 351-385. RANDOM PRODUCTS OF BIRATIONAL MAPS 27

    work page 1983

  28. [28]

    Reine Angew

    Joseph Maher and Giulio Tiozzo,Random walks on weakly hyperbolic groups, J. Reine Angew. Math.742(2018), 187–239

    work page 2018

  29. [29]

    ,Random walks, WPD actions, and the Cremona group, Proc. Lond. Math. Soc. 123.2(2021), 153–202

    work page 2021

  30. [30]

    Arnaud Nerri` ere,Random dynamics of plane polynomial automorphisms, arXiv:2605.02459

    work page internal anchor Pith review Pith/arXiv arXiv

  31. [31]

    Andres Quintero Santander,On the Holomorphic and Random Dynamics for some examples of higher rank Free Groups generated by H´ enon type maps, arXiv:2602.02324 (2026)

    work page arXiv 2026

  32. [32]

    Megan Roda,Classifying hyperbolic ergodic stationary measures on compact complex surfaces with large automorphism group, arXiv:2410.18350 (2024)

    work page arXiv 2024

  33. [33]

    Synth` eses

    Nessim Sibony,Dynamics of rational maps onP k, in Dynamique et g´ eom´ etrie complexes, Panor. Synth` eses. Soc. Math. Fr, 1999

    work page 1999

  34. [34]

    Reine Angew

    Christian Urech,Subgroups of elliptic elements of the Cremona group, J. Reine Angew. Math 770(2021), 27–57

    work page 2021

  35. [35]

    Universit´e Bourgogne Europe, CNRS, IMB UMR 5584, 21000 Dijon, France Email address:arnaud.nerriere@u-bourgogne.fr

    Mingchen Xia,Transcendantal b-divisors I- Correspondence with currents, arXiv:2603.14348 (2026). Universit´e Bourgogne Europe, CNRS, IMB UMR 5584, 21000 Dijon, France Email address:arnaud.nerriere@u-bourgogne.fr

    work page arXiv 2026