arXiv:2607.11421v2

Atomic physical measures for non-invertible random dynamical systems

Vincent P. H. Goverse, Victor Kleptsyn

math.DS37H1237C4037E1037A30

Paper metadata

Original submission: July 13, 2026
Latest revision: July 14, 2026 · version 2
Claimed MSC: 37H12, 37C40, 37E10, 37A30
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 33 pages, 8 figures

Abstract

We construct an example of a random dynamical system on the circle, formed by maps that are only locally invertible, which possesses an atomic stationary measure $ν$ . Moreover, this measure is physical: for Lebesgue-almost every initial point $x_0$ , the Cesàro averages of its random trajectory almost surely converge to $ν$ . This shows that the Hölder regularity of stationary measures, known for (non-measure-preserving) random dynamical systems formed by diffeomorphisms, cannot be generalized to this class of systems. We also provide some related examples, including ones where a stationary measure charges a proper submanifold, despite the absence of a closed common invariant submanifold.

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Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The three substantive clauses of Theorem 1.1 are supported. The constructed generators are smooth local circle diffeomorphisms with no common invariant probability. Their random walk has a unique stationary probability supported on a countable orbit. That measure is physical: empirical measures from Lebesgue-typical initial points converge to it for almost every sample path.

Theorem 1.1Correct

Atomic physical stationary measure for smooth local circle diffeomorphisms

Pages 2–3 and 22–23 · Theorem 1.1 and its proof · arXiv:2607.11421v2

The paper constructs a finite family of smooth, orientation-preserving local diffeomorphisms of the circle whose random walk has a countably atomic stationary probability. This stationary measure is physical: for Lebesgue-almost every initial point and almost every driving sequence, empirical measures converge to it. The maps have no common invariant probability, so the example is not explained by a shared deterministic invariant set. The atoms form the stated countable orbit, and positive recurrence makes their stationary weights summable and unique.

Theorem 1.1, stationary measureCorrect

Positive recurrent countable orbit

Pages 2–3 and 9–14 · Theorem 1.1 and Theorem 3.5 · arXiv:2607.11421v2

On the dyadic orbit, the generators induce a countable Markov chain. The level function has uniformly negative expected drift outside a finite set and bounded increments, so Foster's criterion gives positive recurrence. Irreducibility on the orbit makes the stationary probability unique. Its support is countable and every orbit point has positive mass, establishing the atomic clause of the theorem.

Theorem 1.1, physicalityCorrect

Almost-sure attraction from Lebesgue-typical points

Pages 3 and 18–23 · Proposition 4.7 and final proof · arXiv:2607.11421v2

The depth of a typical point relative to dyadic intervals has an exponential Lebesgue tail, uniformly under the random generators after the level correction. Borel–Cantelli and the ergodic theorem show that empirical measures spend asymptotically full mass in shrinking neighborhoods of the dyadic orbit. Any weak limit is stationary and supported on that orbit, hence equals the unique countable-state stationary law. Compactness then upgrades subsequential identification to convergence of the full empirical sequence.

No common invariant measureCorrect

The generator family has no shared invariant probability

Pages 14–18 · Section 4 and Theorem 4.5 · arXiv:2607.11421v2

Two of the semigroup generators have incompatible attracting/transport behavior on the complementary intervals. Invariance under the first forces mass onto its fixed-point structure, while invariance under the second moves that structure to a disjoint orbit. The resulting equalities force zero total mass, a contradiction. Smooth conjugation preserves this incompatibility, so the final circle maps have no common invariant probability.

02Proofs3 reported findingsCorrect

Theorems 3.5, 4.5 and Proposition 4.7. The initial piecewise-linear Thompson-semigroup action preserves the dyadic orbit. A Foster–Lyapunov function with negative drift outside a finite set makes the induced countable-state chain positive recurrent and yields its unique stationary law. The Ghys–Sergiescu smoothing conjugates the generators to smooth local diffeomorphisms without changing the semigroup relations or orbit chain. For a Lebesgue-typical starting point, level/depth estimates have exponential tails and show that time spent away from arbitrarily small neighborhoods of the dyadic orbit tends to zero. Every empirical limit is therefore supported on that orbit and must equal its unique stationary law.

Drift and smoothing constructionCorrect and complete

Theorems 3.5, 4.5 and Proposition 4.7

Pages 9–23 · Sections 3–4 · arXiv:2607.11421v2

The initial piecewise-linear Thompson-semigroup action preserves the dyadic orbit. A Foster–Lyapunov function with negative drift outside a finite set makes the induced countable-state chain positive recurrent and yields its unique stationary law. The Ghys–Sergiescu smoothing conjugates the generators to smooth local diffeomorphisms without changing the semigroup relations or orbit chain. For a Lebesgue-typical starting point, level/depth estimates have exponential tails and show that time spent away from arbitrarily small neighborhoods of the dyadic orbit tends to zero. Every empirical limit is therefore supported on that orbit and must equal its unique stationary law.

Section 3Correct and complete

Foster–Lyapunov recurrence on dyadic points

Pages 9–14 · Section 3 and Theorem 3.5 · arXiv:2607.11421v2

The level/depth function changes by a bounded amount under each generator. Averaging over the driving law gives a strict negative drift beyond a finite core. The chain is irreducible on the dyadic orbit and the finite core is petite, so the Foster criterion yields positive recurrence and a unique stationary probability. Return-time integrability supplies the ergodic theorem used later.

Section 4Correct and complete

Smooth realization and empirical-limit identification

Pages 14–23 · Section 4 · arXiv:2607.11421v2

The Ghys–Sergiescu conjugacy smooths the Thompson generators while preserving all algebraic relations and maps the dyadic orbit to a countable smooth orbit. Distortion estimates compare interval depth before and after each random word and give the exponential tail in Proposition 4.7. Tightness holds on the compact circle. Stationarity of empirical limits follows from the vanishing endpoint discrepancy, and support concentration plus uniqueness on the orbit identifies every limit.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.11421v2
Authors listed: Vincent P. H. Goverse, Victor Kleptsyn
Audit date: August 18, 2026

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Atomic physical measures for non-invertible random dynamical systems

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Atomic physical stationary measure for smooth local circle diffeomorphisms

Positive recurrent countable orbit

Almost-sure attraction from Lebesgue-typical points

The generator family has no shared invariant probability

Theorems 3.5, 4.5 and Proposition 4.7

Foster–Lyapunov recurrence on dyadic points

Smooth realization and empirical-limit identification

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