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Convolution semigroups for automorphism dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper defines a convolution product on invariant types and Keisler measures over arbitrary first-order theories and proves that, in NIP and stable settings, idempotent fim measures correspond exactly to unique invariant measures on…

desk verdict Visible part is strong, original work; the stable-theory headline is announced but not present in the supplied text, so review should be conditional until the full manuscript is available. read the letter →

arxiv 2507.23503 v1 pith:YWZRC4N3 submitted 2025-07-31 math.LO

Kyle Gannon , Daniel Max Hoffmann , Krzysztof Krupiński This is my paper
classification math.LO MSC 03C4503C9537B0228C1028E1543A05
keywords modeltheoryKeislermeasuresconvolutionEllissemigroupautomorphismgroupNIPtheoriesstableidempotent
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

Working with the automorphism group of a saturated 'monster' model in place of a definable group, the paper builds a convolution operation on global invariant types and Keisler measures. In NIP theories it shows the Ellis semigroup of the natural measure flow is homeomorphic to the space of strongly finitely satisfiable measures, so the semigroup product can be transferred and then extended by an integral formula to all Borel-definable invariant measures. The central classification conjecture is that a fim measure is idempotent exactly when it is the unique invariant measure concentrated on its own stabilizer subgroup; this gives a correspondence between idempotent fim measures and relatively type-definable fim subgroups of the automorphism group. The conjecture is proved for generically stable types in rosy theories, for Kim-Pillay-invariant measures in NIP theories, and for all fim measures in stable theories, while a general NIP version remains conditional on an open associativity question.

What carries the argument

The load-bearing mechanism is the pair $\Phi : E(M_{\bar x}(M), \mathrm{conv}(G_{\pi,M})) \to M^{sfs}_{\pi(\bar m;\bar y)}(C,M)$ and its inverse $\Psi$, which identify Ellis-semigroup composition with the integral convolution $(\mu * \nu)(\varphi(\bar b;\bar y)) = \int_{S_{\bar m}(C)} G^{\varphi(\bar b;\bar y)}_{\mu}\,d\nu$. Around this sit the relatively $\bar m$-type-definable subgroups $G_{\pi,C} = \{\sigma \in \mathrm{Aut}(C) : \models \pi(\sigma(\bar m);\bar m)\}$, the stabilizer $\mathrm{Stab}(\mu)$, the uniqueness lemma (Lemma 6.12) for fim measures, and Theorem 7.25, a counterpart of Newelski's group chunk theorem for $\mathrm{Aut}(C)$ in stable theories.

What would settle it

Construct an idempotent fim measure $\mu$ whose stabilizer subgroup $\mathrm{Stab}(\mu)$ carries two distinct left-invariant Borel-definable measures; Conjecture (A) would then fail. Alternatively, find a triple of $M$-invariant measures in an NIP theory for which $\mu * (\nu * \lambda)$ differs from $(\mu * \nu) * \lambda$, resolving Question 1.1 negatively.

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Extended reading notes

Core claim

The paper's central discovery is a transfer result: when $T$ is NIP and a partial type $\pi$ is group-like over $M$, the maps $\Phi$ and $\Psi$ give a homeomorphism between the Ellis semigroup of the flow $(M_{\bar x}(M), \mathrm{conv}(G_{\pi,M}))$ and the space $M^{sfs}_{\pi(\bar m;\bar y)}(C,M)$ of strongly finitely satisfiable measures concentrating on $\pi(\bar m;\bar y)$. Through this homeomorphism the composition product of the Ellis semigroup becomes an explicit convolution $\mu * \nu$, defined by integrating the fiber function $F^{\varphi}_{\mu}$ against the pushforward of $\nu$ under the maps $h_{\bar b}$. This convolution is then defined for arbitrary Borel-definable invariant measures, and the authors prove Conjecture (A) in several settings: an idempotent fim measure is precisely the unique left invariant measure on its stabilizer subgroup. The main announced result of the second part is confirmation of Conjecture (A) for all fim measures in stable theories, obtained via a group-chunk theorem for the automorphism group of the monster model.

Load-bearing premise

The classification of idempotent measures relies on Lemma 6.12, which requires a fim measure to be super-fim, the theory to be NIP, or the language and model to be countable; outside those cases the uniqueness half of Conjecture (A) is not proved, and in full NIP associativity of the convolution is still open.

Editorial extensions

If this is right

  • In NIP theories, $(M^{fs}_{\bar m}(C,M), *)$ is a compact left topological semigroup, and when the theory and model are countable the same holds for the full space of invariant measures.
  • The convolution is associative on invariant types in every theory, on definable measures without NIP, and on finitely satisfiable measures under NIP; general NIP associativity remains open.
  • Conjecture (A) holds for generically stable types in rosy theories, for stable types, for Kim-Pillay-invariant measures in NIP, and for all fim measures in stable theories, so in these settings idempotent fim measures correspond exactly to relatively type-definable fim subgroups of $\mathrm{Aut}(C)$.
  • Through the affine sort construction, the new product specializes to the definable-group convolution, so the paper's classifications imply the corresponding definable-group results in the recent literature.
  • The transfer homeomorphism supplies explicit examples: an idempotent definable measure in the random graph that is not finitely satisfiable, an idempotent generically stable type in a blow-up structure, and a generically stable idempotent average in a stable theory.

Reading between the lines

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

  • One testable extension is to check whether, once associativity is settled, the convolution product turns $M^{inv}_{\bar m}(C,M)$ into a compact left topological semigroup in all NIP theories; the paper only achieves this under countability assumptions.
  • The authors' proposed definable analogues of Martin and Poisson boundaries would make this convolution the transition kernel of a definable random walk, and the semigroup structure established here is the natural first step.
  • The stable-theoretic group chunk theorem for $\mathrm{Aut}(C)$ may be applicable beyond idempotent measures, for instance to classify stabilizers of generic types or to study relatively definable subgroups in stable theories.
  • A concrete experiment is to use the affine sort construction to transfer a hypothetical counterexample to the definable-group conjecture back to Conjecture (A), yielding a direct test of the paper's main correspondence.
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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 develops a convolution product for invariant Keisler measures and types over arbitrary first-order theories, transferring Ellis semigroup operations from automorphism-group flows to spaces of strongly finitely satisfiable types and measures. Under NIP, Theorem 4.9 gives a homeomorphism between the Ellis semigroup of the affine action on measures and the space of strongly finitely satisfiable measures, and Definition 4.21 extends the induced product to a convolution on all M-invariant Borel-definable measures. The second part formulates Conjecture (A), identifying idempotent fim measures with unique invariant measures on their stabilizers, and proves it in several visible settings: types in rosy theories and stable types (Corollary 6.21), Kim-Pillay invariant measures in NIP (Theorem 6.31), and—according to the introduction—all fim measures in stable theories (Corollary 8.25). The supplied manuscript, however, breaks off in the middle of Section 7.1, before Theorem 7.25 and Sections 8.1–8.3 are presented, so the announced stable classification is not verifiable from the text provided.

Significance. If the complete manuscript delivers the announced results, this is a substantial contribution: it introduces a natural convolution operation for invariant Keisler measures in arbitrary theories, connects it explicitly to Ellis semigroups and to Hrushovski-Newelski dynamics, and proves a broad correspondence between idempotent measures and relatively type-definable subgroups of automorphism groups. The visible portions are strong and contain real achievements: an explicit homeomorphism theorem (Theorem 4.9), an integral formula for the convolution (Proposition 4.20 and Definition 4.21), several associativity results on important subspaces (Propositions 4.33, 4.35, 4.36, 4.37, 4.40), the affine-sort encoding of definable-group convolution (Theorem 5.23 and Corollary 5.24), and the Kim-Pillay invariant classification in NIP (Theorem 6.31). The paper is also honest about its limitations, explicitly recording that full associativity in NIP and the super-fim question remain open. However, the flagship stable case, Corollary 8.25, rests on arguments that are not visible in the supplied text, so the overall significance can only be evaluated conditionally.

major comments (3)
  1. [§1.3, announced Corollary 8.25] The paper's announced main result, Corollary 8.25 (Conjecture (A) for all fim measures in stable theories), is not established in the text submitted for review. The introduction states that it follows from Theorem 7.25, a counterpart of Newelski's theorem, but the supplied manuscript stops in the middle of Section 7.1, before Theorem 7.25 is stated or proved and before Sections 8.1–8.3 appear. I therefore cannot verify the central stable classification. This is a missing-proof concern rather than an identified contradiction, but it is load-bearing for the paper's main claim.
  2. [§7.1, toward Theorem 7.25] The visible part of Section 7.1 develops genericity and R_Delta-rank criteria for relatively definable subsets of G_{π,C} (Propositions 7.3, 7.7, and 7.9), but the announced group-chunk step—that the smallest relatively ¯m-type-definable over M subgroup containing a given relatively invariant set exists and is generated by generics—is not visible. If Theorem 7.25 or the Section 8 arguments require extra hypotheses (for example, that the stabilizer is fim, or a uniqueness statement beyond Lemma 6.12), Corollary 8.25 would not follow as stated. The full proof of Theorem 7.25 and its application in Section 8 must be supplied before the stable classification can be assessed.
  3. [§6.2, Lemma 6.12, Corollary 6.13(3), Question 2.14] The visible uniqueness results for invariant measures are conditional: Lemma 6.12 assumes super-fim, countability, or NIP, and Question 2.14 records that it is open whether every fim measure is super-fim. Consequently, Corollary 6.13(3) establishes the 'unique invariant measure' half of Conjecture (A) only under an additional hypothesis. The stable case is supposed to remove this limitation via the missing Section 8, but since that part is not available, the full conjecture is not settled by the visible portion of the paper. The text states this limitation honestly, but it remains a substantive gap in the submitted version.
minor comments (5)
  1. [Throughout] There are several typographical errors that should be corrected in revision: 'homeomorhism' in the introduction, 'Morely product' in the proof of Theorem 4.9, 'ηη' in the proof of Lemma 4.5, and 'Asumme' in Theorem 4.40.
  2. [§7.1] The change of convention for ¯x (from variables enumerating M to variables enumerating C) is explicitly flagged, but it is a serious notational overload. Using a different symbol, such as ¯z, for the monster-model enumeration would make Section 7 substantially easier to read.
  3. [Question 1.1 and §4.2.3] Because full associativity of the convolution product in NIP is left open, the word 'semigroup' is used in a qualified sense. The paper is careful about this, but a short remark at the start of Section 4.2.3 clarifying that all later idempotence statements use the binary product directly, independently of associativity, would help avoid confusion.
  4. [Proposition 4.38] The proof of Proposition 4.38 is deferred to 'a similar argument from [CG23]'. This is acceptable, but a one-sentence indication of how the triviality of the Ellis group is obtained would make the paper more self-contained.
  5. [Example 6.1] In the random graph example, the notation R^1 and R^0 for the edge relation and its negation should be stated explicitly, since the formula uses both superscripts and subscripts and the current presentation is easy to misread.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the convolution product is defined by an explicit integral formula, and the Ellis-semigroup transfer is used as a construction rather than as an unsupported prediction.

full rationale

The paper's derivation chain is not circular. The convolution product on strongly finitely satisfiable measures is first obtained by transferring the Ellis semigroup product through the homeomorphism of Theorem 4.9, but the product is then given independently by the explicit integral formula in Definition 4.21 and Proposition 4.20, which is shown to extend the transferred product. The classification results compare idempotence against stabilizer subgroups defined independently in Definition 2.19 and Lemma 2.26, with uniqueness established through the separately proved Lemmas 6.11 and 6.12, whose assumptions (super-fim, NIP, or countability) are stated and not assumed from the conjecture. The affine-sort transfer in Section 5 is an explicit isomorphism argument, not a renaming. Some cited background results come from the authors' prior work, but they are used as published lemmas about Morley products and definable-group convolution, not as an unverified uniqueness theorem that already contains Conjecture (A); they therefore constitute independent support rather than a self-citation chain. The announced stable-theoretic classification (Corollary 8.25) lies outside the supplied text, but the visible material up to Section 7.1 shows no step in which a claimed output is definitionally identified with an input. Accordingly, no significant circularity is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No free parameters or fitted constants appear; the paper is pure mathematics. The load-bearing axioms are standard set-theoretic and model-theoretic background (monster models, compactness), the ambient tameness hypotheses (NIP, stability, rosy), the group-like homogeneity condition, and the background theorems on definable convolution and Morley products, mostly from published works by the same authors. The genuinely new objects, the convolution product and the sfs subspaces, are each anchored to independent external structures, the Ellis semigroup and the definable convolution semigroup.

assumptions (6)
  • standard math The monster model C is kappa-saturated and strongly kappa-homogeneous for a strong limit cardinal kappa, with C' a larger monster in which C is small.
    Standard model-theoretic background assumed at the start of Section 2.3 and used throughout, for example to represent every p in S_mbar(C) as tp(sigma(mbar)/C) for some sigma in Aut(C').
  • domain assumption NIP: every global M-invariant Keisler measure is Borel-definable over M (Fact 2.9(1), cited to [HP11]), and Morley products are associative and left continuous on invariant measures under NIP (Fact 2.9(3)-(4), cited to [CG21; CG22]).
    This is the ambient hypothesis for the inverse map Psi, for the homeomorphism Theorem 4.9, and for the measure-level convolution theory in Section 4.2 and the NIP classification results in Section 6.
  • domain assumption The pair (M, pi) is group-like (Definition 4.1): pi(xbar;ybar) proves xbar =_empty ybar, G_pi,M is a subgroup, and every finite partial match of pi-realizing tuples is realized by an element of G_pi,M.
    Needed for Phi to be well-defined (Lemma 4.5) and for Psi to be well-defined via the approximation of sfs measures by convex combinations of delta_{sigma(mbar)} (Lemma 4.8). For pi equal to 'xbar =_empty ybar' it reduces to strong aleph0-homogeneity of M.
  • domain assumption Stability in Sections 7 and 8: all R_Delta-ranks are finite, non-forking extensions are definable, and every invariant measure is definable.
    Used for Proposition 7.3 and Proposition 7.9 (genericity versus non-forking), for the Newelski-type theorem Theorem 7.25, and for the stable classification Corollary 8.25; the review copy truncates during Section 7.1, so the full derivation of the stability machinery was not visible.
  • domain assumption In Lemma 6.12 and Corollary 6.13, the fim measure is either super-fim, or the language and model are countable, or the theory is NIP (in which case fim implies super-fim, cited to [HPS13]).
    The equality mu = nu from mu|M = nu|M for Borel-definable fim measures requires one of these hypotheses; Question 2.14 flags that super-fim in general is open, so the uniqueness of left G_pi,C-invariant measures is conditional outside NIP and countable settings.
  • domain assumption Background theorems on definable convolution from [CG22], [CG23], [CGK24], and the affine sort facts from [GN08] (Fact 5.5) and [KR16] are correct.
    Section 5 uses these to prove the transfer of properties (Corollary 5.19) and the semigroup isomorphism (Theorem 5.23) that shows the new convolution encodes definable convolution; these are published results by overlapping authors.
invented entities (2)
  • The convolution product * on invariant Keisler measures and types over arbitrary theories (Definition 4.21) independent evidence
    purpose: Defines harmonic analysis and random automorphisms for arbitrary first-order theories, and is the operation whose idempotents are classified by subgroups of Aut(C).
    The product coincides with the Ellis semigroup transfer on the sfs subspaces (Theorem 4.9) and reduces to the definable-group convolution via the affine sort (Theorem 5.23), giving two independent anchors outside the paper itself.
  • Spaces of strongly finitely satisfiable types and measures S^sfs and M^sfs (Definition 2.16) independent evidence
    purpose: Serve as the measure-theoretic targets of the Ellis semigroup homeomorphisms and as the domain where the transferred product is associative by construction.
    These are closed subspaces of the standard type and measure spaces, characterized by an explicit finite-satisfiability condition; for pi equal to 'x =_empty y' and strongly homogeneous M they coincide with the usual finitely satisfiable types and measures (Remark 6.23).

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Cite this review

Pith. "Pith review of Convolution semigroups for automorphism dynamics." pith.science (2026). https://pith.science/paper/YWZRC4N3

@misc{pith2026250723503,
  author       = {Pith},
  title        = {Pith review of: Convolution semigroups for automorphism dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YWZRC4N3}},
  note         = {Machine review of arXiv:2507.23503}
}
read the original abstract

Initially motivated by Hrushovski's paper on definability patterns, we obtain homeomorphisms between Ellis semigroups related to natural actions of the automorphism groups of first order structures and certain collections of types and Keisler measures. Thus, we can transfer the semigroup operation from these Ellis semigroups to the corresponding collections of types and Keisler measures. By generalizing this transferred product, we obtain a new convolution operation for invariant types and measures in arbitrary first-order theories. We develop its general theory and prove several correspondence theorems between idempotent measures and closed subgroups of the automorphism group of a sufficiently large (so-called monster) model with respect to the relatively definable topology. Via the affine sort construction, we demonstrate that this new notion of convolution encodes the standard definable convolution operation over definable groups.

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