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Attracting measures

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Pith's one-line read The paper proves that a mixing invariant measure on an orbit-tracking attractor is attracting whenever a uniform tube-mass ratio is bounded below—so every absolutely continuous ensemble started near the attractor converges to it.

desk verdict A correct and honest topological abstraction of the Bowen–Ruelle argument that mixing SRB measures are attracting; the main theorem is conditional on a 3δ tube-mass estimate that is not established, and the paper says so. read the letter →

arxiv 2506.02136 v1 pith:MHIQBMDS submitted 2025-06-02 math.DS

Julian Newman , Peter Ashwin This is my paper
classification math.DS MSC 37A0537A2537C7037D20
keywords attractingmeasurephysicalSRBmixingorbit-trackingattractorensembleconvergenceAxiomAtube-massestimate
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper singles out a second statistical naturalness property of invariant measures—called attracting—alongside the usual physical-measure property, and argues it deserves attention in its own right. A measure is attracting when every absolutely continuous probability distribution released near its support is carried by the dynamics to that measure in the limit, a property that is experimentally accessible by simulating ensembles of trajectories. The main theorem gives a topological abstraction of the classical argument that SRB measures on Axiom A attractors are attracting: a mixing measure whose support is an orbit-tracking attractor is attracting provided a uniform lower bound on a tube-mass ratio holds. The paper also constructs a smooth flow whose invariant measure is attracting but neither physical nor ergodic, proving that the new notion is not a consequence of the familiar ones.

What carries the argument

The argument is carried by three ingredients. Mixing measures (Definition 10) have the property that every $\mu$-absolutely continuous probability measure evolves weakly to $\mu$, so once the proof replaces the initial absolutely continuous ensemble by a $\mu$-absolutely continuous approximation, mixing does the rest. Attractors via orbit-tracking (Definitions 23–25) are invariant compact sets for which a fixed neighbourhood eventually shadows every point trajectory by a trajectory on the set, uniformly in the starting point. The tube-mass estimate (2) is the uniform lower bound, over $x\in A$ and an unbounded set of time horizons, of $\mu(\{y\in A:\, d(f^t x, f^t y)<\delta \text{ for all } t\in[0,\tau]\})$ divided by $m(\{y\in X:\, d(f^t x, f^t y)<3\delta \text{ for all } t\in[0,\tau]\})$; it is the quantitative 'chaoticness is no greater under $\mu$ than under Lebesgue measure' ingredient. The proof then uses $(d,\delta)$-bi-separated subsets of $A$ (Lemma 30) to cut the tube around $A$ into finitely many pieces, builds $\mu$-absolutely continuous approximations $P[\nu;\mu,M,N]$ (Definition 31) by redistributing the transported ensemble over $\mu$-mass on matching pieces, and applies dominated weak convergence (Lemma 33) to preserve absolute continuity in the limit.

What would settle it

A decisive check on the reach of Theorem 27 is to find a system that satisfies all three hypotheses—a mixing invariant measure $\mu$, an orbit-tracking attractor $A=\operatorname{supp}\mu$, and the tube-mass bound (2)—but for which some absolutely continuous probability measure supported near $A$ fails to converge weakly to $\mu$; such a system would refute the theorem. Short of that, a concrete numerical surrogate is to estimate, for a candidate attractor such as a non-uniformly hyperbolic one, the ratio $\mu(B^\tau_\delta(x))/m(B^\tau_{3\delta}(x))$ over long time horizons and many base points: if the infimum tends to zero for arbitrarily small $\delta$, the theorem's main hypothesis is not met, and the question of whether the measure is attracting must be decided by direct ensemble simulation.

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Extended reading notes

Core claim

On the paper's own terms, the core discovery is Theorem 27. A measure $\mu$ is called attracting (Definition 12) if some open neighbourhood $U$ of its support has the property that $f^t \nu \to \mu$ weakly for every reference-measure-absolutely-continuous probability measure $\nu$ with $\nu(U)=1$; this is the ensemble analogue of a physical measure, whose typical single orbits produce the right empirical statistics. The theorem states that if $\mu$ is mixing, its support $A$ is an attractor via orbit-tracking, and for arbitrarily small $\delta$ there is an unbounded set of time horizons $\mathcal{T}$ such that the $\mu$-mass of a $\delta$-tube around a segment of an $A$-orbit divided by the $m$-mass of the corresponding $3\delta$-tube in the ambient space is uniformly bounded below, then $\mu$ is attracting. This reproduces, at a purely topological and measure-theoretic level, the mechanism by which SRB measures on Axiom A attractors were known to be attracting, and it identifies the three properties—mixing, orbit-tracking, and controlled chaoticness relative to the reference measure—that do the work.

Load-bearing premise

The argument collapses if the uniform tube-mass estimate (2) fails: the paper does not prove this estimate for any system beyond the Axiom A case it cites, so the advertised extension stands or falls on future verification that the $\mu$-mass of thin tubes near the attractor is never negligible compared with the $m$-mass of wider ambient tubes.

Editorial extensions

If this is right

  • Any mixing invariant measure on an orbit-tracking attractor satisfying the tube-mass bound is attracting: releasing any absolutely continuous ensemble in a neighbourhood of the attractor yields weak convergence to $\mu$.
  • Along with Proposition 20, this means the convergence holds for every absolutely continuous probability measure supported in the basin of the attractor, not just in one chosen neighbourhood.
  • The theorem isolates the minimal ingredients behind the classical Axiom A result, so verifying the same three properties for other systems—for example non-uniformly hyperbolic attractors—would automatically give ensemble naturalness there.
  • Because every mixing measure is ergodic, the measures admitted by Theorem 27 are both ergodic and attracting, while the Proposition 21 example shows that attracting measures can fail to be ergodic or physical.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not verify the tube-mass estimate (2) for any concrete system beyond the Axiom A setting it cites; checking it numerically or analytically for a non-uniformly hyperbolic attractor would be the decisive next experiment.
  • The proof's use of $3\delta$ rather than the $2\delta$ in the classical argument suggests the factor is an artifact of the bi-separation covering, not a sharp constant; one might conjecture that the same theorem holds with any $c>1$ in the denominator, though the paper does not address this.
  • Because the outlook connects attracting measures of an autonomous past limit to ensemble behaviour under parameter drift, Theorem 27 gives a potential bridge: an autonomous system satisfying the theorem would automatically supply the attracting past-limit measure needed for the nonautonomous tipping analysis, without re-doing the SRB construction.
  • The terminology 'attracting' invites a shift in numerical practice: for singular invariant measures, ensemble convergence is easier to measure than decay of correlations, so the theorem's hypotheses give a practical criterion for when repeated simulations from random initial data should converge to a single invariant law.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces the notion of an 'attracting measure' as a compactly supported invariant measure μ for which every Lebesgue-absolutely continuous probability distribution initially supported near supp μ is weakly pushed forward to μ by the dynamics. This formalizes the second naturalness property of SRB measures on Axiom A attractors, alongside the more familiar 'physical measure' property. The paper develops basic theory: basin characterizations, implications and non-implications among ergodic, physical, mixing and attracting measures, and a C^1 flow example (Proposition 21) showing that attracting measures need not be physical or ergodic. The main result, Theorem 27, is a topological abstraction of the Bowen–Ruelle strategy: if μ is mixing, its support is an attractor via orbit-tracking, and a uniform tube-mass estimate (condition (2)) holds, then μ is attracting. The proof is a careful, self-contained reworking using bi-separated sets, approximation by μ-absolutely continuous measures, and dominated weak convergence.

Significance. The proof of Theorem 27 is detailed and appears correct; the lemmas (Lemma 1, Lemma 30, Lemma 32, Lemma 33) are cleanly stated, and the key hypothesis is isolated in condition (2). If that condition can be verified for a system, the theorem immediately gives a strong ensemble-based naturalness property of the type highlighted in the abstract. The paper also contributes the useful observation that attracting measures are distinct from physical and mixing measures, and it provides a conceptual framework that may guide future work on non-Axiom-A attractors. However, the central theorem is conditional: condition (2) is not verified for any nontrivial example, and the 3δ tube-mass estimate needed by the proof is not presently established for the motivating Axiom A SRB case, for which only a 2δ estimate is cited. The paper's significance therefore rests on future verification of (2); this is acknowledged in the Outlook but should be reflected more explicitly in the presentation of the main theorem.

major comments (3)
  1. [Sec. 2.4, Proposition 20(B)] Proposition 20(B) is false as stated. Let X={p,q} with both points fixed and m the counting measure. The measure μ=(δ_p+δ_q)/2 is invariant, its support A={p,q} is an attractor, and Basin(μ)=∅. The condition 'for every m-absolutely continuous probability measure ν with ν(Basin(μ))=1, f^tν→μ' is thus vacuously true, yet μ is not attracting because f^tν=ν for every ν and ν does not converge to μ unless ν=μ. The proof actually establishes the statement with Basin(suppμ) in place of Basin(μ); please correct the statement and the conclusion of the proof accordingly.
  2. [Sec. 3.2, Theorem 27 and condition (2)] The advertised connection to SRB measures on Axiom A attractors is not established. Condition (2) uses a 3δ tube mass in the denominator, while the cited result [3, Corollary 4.6] provides a 2δ estimate. The covering argument in the proof of Theorem 27 (via Lemma 30 with (dτ,δ)-bi-separated sets) forces the 3δ denominator, and no comparison m(B_{3δ}(x,τ)) ≤ K m(B_{2δ}(x,τ)) is proved or referenced. Thus, as it stands, Theorem 27 is a conditional statement with no verified nontrivial instance; the Outlook's question of whether it applies beyond Axiom A remains open. Please either supply a 3δ estimate for the motivating class or clearly frame the theorem as conditional on an unverified hypothesis.
  3. [Sec. 3.2, Remark 28] Remark 28 asserts that the proof of [3, Theorem 5.3] 'does not then seem to go through correctly' and claims the need for a τ-independent bound on the intersection multiplicity of (dτ,δ)-balls. This is a strong assertion about a classical proof and is not substantiated in the text. Please expand the remark into a rigorous explanation—with a precise statement of the step that fails and the missing bound—or soften the claim so that it does not appear as an unproved criticism of a published result.
minor comments (5)
  1. [Sec. 1.1, definition of subcone] The definition of a subcone reads 'for every h∈L1+(m) and c>0, ch∈L1+(m)', which is nonsensical; it should read 'for every h∈C and c>0, ch∈C'.
  2. [Proof of Theorem 27] The real number T used as a time shift is denoted by the same symbol as the unbounded set T in condition (2); please use different notation (e.g., T0) to avoid confusion.
  3. [Abstract] The abstract says that the second property holds 'under mild assumptions', but in the flow case an extra condition (e.g., dense unstable manifolds) is required; consider making this qualification explicit in the abstract.
  4. [Proof of Proposition 20(B)] The density of the additive semigroup generated by the set C in L^1_+(m|Basin(suppμ)) is not immediate from the text; a short justification (e.g., via a partition of unity subordinate to the open cover {f^{-n}U}) would improve the proof.
  5. [Proposition 21] The formula for the vector field b is hard to parse; a displayed equation with explicit coordinates would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 27 is a conditional abstraction; the unproved tube estimate is an external-support gap, not a self-referential reduction.

full rationale

The paper's central result, Theorem 27, is a conditional statement: it assumes that the measure is mixing, that its support is an attractor via orbit-tracking, and that a uniform tube-mass estimate (2) holds, and it proves that the measure is attracting. The conclusion is not assumed in the hypotheses, and condition (2) is not a restatement of the attracting-measure definition: attracting requires convergence of f^t nu to mu for every absolutely continuous initial distribution in a neighbourhood, while (2) is a lower bound on mu of a thin dynamical tube divided by the Lebesgue measure of a wider tube. The proof uses (2) to control the densities d(mu_n)/d(mu), applies Lemma 33 to get mu-absolute continuity of a weak limit, and then uses mixing to finish; this is a genuine derivation rather than an identity in disguise. There are no fitted parameters and no prediction that is statistically forced by construction. The self-citations [1,7] are used only to say that the terminology 'attracting measures' was introduced there, and that the authors' prior work on asymptotically autonomous systems motivates the concept; this is not load-bearing for the theorem. The external citation to Bowen and Ruelle [3] supplies the SRB facts: the mixing property and, in Corollary 4.6, the 2-delta version of the tube estimate. Remark 28 explicitly notes that the present proof requires 3-delta instead of 2-delta and that the corresponding step in [3, Theorem 5.3] 'does not then seem to go through correctly' absent an additional bound. This is an honest admission of an unresolved step and a gap in the external support for applying the theorem to Axiom A SRB measures, but a gap or correctness risk is not circularity. The theorem remains a valid conditional result whose advertised reach depends on future verification of (2) or a 3-delta comparison; that is an open-support issue, not a self-referential derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard measure theory, a general structural assumption (H), and the external tube-mass estimate (2) for SRB measures. No free parameters are fitted, and no new physical or mathematical entities are postulated.

assumptions (5)
  • domain assumption General setup (H): the reference measure m is locally finite with full support, and X can be covered by open sets U such that for each t>=0 the map S -> m(U intersect f^{-t}S) is m-absolutely continuous with bounded density.
    Assumed in Section 2.1 to ensure pushforwards of absolutely continuous measures remain absolutely continuous, as needed for Lemmas 2 and 3.
  • standard math The semiflow (f^t)_{t>=0} is continuous in (t,x) and X is a separable metric space.
    Background topological conditions used throughout for weak convergence and measure-theoretic arguments.
  • domain assumption Condition (2): for arbitrarily small delta, there is an unbounded set T such that the infimum over x in A and tau in T of mu(delta-tube in A)/m(3delta-tube in X) is positive.
    This is the controlled-chaoticness hypothesis of Theorem 27. For SRB measures it is imported from [3, Cor 4.6] with 2delta in place of 3delta, and is not proved in this paper.
  • domain assumption The measure mu is mixing and its support A is an attractor via orbit-tracking.
    Explicit hypotheses of Theorem 27; for Axiom A attractors, orbit-tracking is attributed to [5, Theorem 7.4] and [3, Proposition 4.4].
  • domain assumption Axiom A attractors are attractors via orbit-tracking.
    Used to connect Theorem 27 to the classical SRB setting; cited to [5] and [3] rather than proved in this paper.

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Pith. "Pith review of Attracting measures." pith.science (2026). https://pith.science/paper/MHIQBMDS

@misc{pith2026250602136,
  author       = {Pith},
  title        = {Pith review of: Attracting measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MHIQBMDS}},
  note         = {Machine review of arXiv:2506.02136}
}
abstract

Under mild assumptions, the SRB measure $\mu$ associated to an Axiom A attractor $A$ has the following properties: (i) the empirical measure starting at a typical point near $A$ converges weakly to $\mu$; (ii) the pushforward of any Lebesgue-absolutely continuous probability measure supported near $A$ converges weakly to $\mu$. In general, a measure with the first property is called a "physical measure", and physical measures are recognised as generally important in their own right. In this paper, we highlight the second property as also important in its own right, and we prove a result that serves as a topological abstraction of the original result that establishes the second property for SRB measures on Axiom A attractors.

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Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    Physical invariant measures and tipping probabilities for chaotic attractors of asymptotically autonomous systems.The European Physical Journal Special Topics, pages 1–14, 2021

    Peter Ashwin and Julian Newman. Physical invariant measures and tipping probabilities for chaotic attractors of asymptotically autonomous systems.The European Physical Journal Special Topics, pages 1–14, 2021

    work page 2021

  2. [2]

    Almost sure rates of mixing for i.i.d

    Viviane Baladi, Michael Benedicks, and V´ eronique Maume-Deschamps. Almost sure rates of mixing for i.i.d. unimodal maps.Annales scientifiques de l’ ´Ecole Normale Sup´ erieure, Ser. 4, 35(1):77–126, 2002

    work page 2002

  3. [3]

    The ergodic theory of AxiomAflows.Inventiones mathematicae, 29:181–202, 1975

    Rufus Bowen and David Ruelle. The ergodic theory of AxiomAflows.Inventiones mathematicae, 29:181–202, 1975

    work page 1975

  4. [4]

    Driver.Analysis Tools with Applications

    Bruce K. Driver.Analysis Tools with Applications. Springer, 2003

    work page 2003

  5. [5]

    Hirsch, J

    M. Hirsch, J. Palis, C. Pugh, and M. Shub. Neighborhoods of hyperbolic sets.Inventiones mathematicae, 9:121–134, 1970. 16

    work page 1970

  6. [6]

    V´ asquez

    Esteban Mu˜ noz-Young, Andr´ es Navas, Enrique Pujals, and Carlos H. V´ asquez. A continuous Bowen-Mane type phenomenon.Discrete & Continuous Dynamical Systems - A, 20(3):713– 724, 2007

    work page 2007

  7. [7]

    Physical measures of asymptotically autonomous dynamical systems.Stochastics and Dynamics, 23(03):2350020, 2023

    Julian Newman and Peter Ashwin. Physical measures of asymptotically autonomous dynamical systems.Stochastics and Dynamics, 23(03):2350020, 2023

    work page 2023

  8. [8]

    Parthasarathy.Probability Measures on Metric Spaces

    K.R. Parthasarathy.Probability Measures on Metric Spaces. AMS Chelsea Publishing Series. AMS Chelsea Pub., 2005

    work page 2005

Show all 13 references
  1. [9]

    Ergodic attractors.Transactions of the American Mathematical Society, 312(1):1–54, 1989

    Charles Pugh and Michael Shub. Ergodic attractors.Transactions of the American Mathematical Society, 312(1):1–54, 1989

    1989

  2. [10]

    A measure associated with Axiom-A attractors.American Journal of Mathematics, 98(3):619–654, 1976

    David Ruelle. A measure associated with Axiom-A attractors.American Journal of Mathematics, 98(3):619–654, 1976

    1976

  3. [11]

    Cambridge Studies in Advanced Mathematics

    Marcelo Viana and Krerley Oliveira.Foundations of Ergodic Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2016

    2016

  4. [12]

    What are SRB measures, and which dynamical systems have them? Journal of Statistical Physics, 108(5–6):733–754, 2002

    Lai-Sang Young. What are SRB measures, and which dynamical systems have them? Journal of Statistical Physics, 108(5–6):733–754, 2002

    2002

  5. [13]

    Generalizations of SRB measures to nonautonomous, random and infinite dimensional systems.Journal of Statistical Physics, 166:4924–515, 2016

    Lai-Sang Young. Generalizations of SRB measures to nonautonomous, random and infinite dimensional systems.Journal of Statistical Physics, 166:4924–515, 2016. 17

    2016

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