Ergodic Optimization with Linear Constraints

Kevin McGoff; Shengwen Guo

REVIEW 3 major objections 4 minor 49 references

Ergodic Optimization with Linear Constraints

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Ergodic optimization with linear constraints has unique optimizers for typical objectives, plus a dual formula.

desk verdict A useful unifying framework for constrained ergodic optimization; the main uniqueness theorem is sound, but the advertised scope of the realization theorem is too wide and one key lemma is unproved. read the letter →

arxiv 2608.02435 v1 pith:OTXZAFLJ submitted 2026-08-03 math.DS

Shengwen Guo , Kevin McGoff This is my paper

classification math.DS MSC 37A0537D2049Q2049N15

keywords ergodicoptimizationlinearconstraintsinvariantmeasuresgenericuniquenessprevalencedualityoptimaltransportrealization

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

Ergodic optimization looks for the invariant probability measure that maximizes the average of a given continuous function. This paper adds linear constraints—the measure must give zero average to every function in a specified family—and studies the same maximization over the resulting feasible set. The central claim is that whenever the constrained feasible set is non-empty, a topologically generic and measure-theoretically prevalent set of objective functions has a unique maximizing measure. The paper also gives a precise non-emptiness criterion, a realization result describing which subsets of the feasible set can be optimal, and a duality formula that unifies classical ergodic optimization duality with optimal-transport duality.

What carries the argument

The central object is the feasible set M_C(X,T), defined as the intersection of the compact convex simplex M(X,T) of invariant measures with the common zero sets of the linear conditions ∫g dμ = 0. The main mechanism is the solution-set correspondence that assigns to each objective φ the non-empty set of its maximizers. This correspondence has a closed graph, so by a standard semicontinuity theorem its continuity points form a residual set; continuity of the correspondence is shown to be equivalent to uniqueness of the maximizer. For the measure-theoretic statement, the paper studies the maximal-value functional F_C(φ) = sup∫φ dμ, which is Lipschitz; its points of directional differentiabili

What would settle it

Find a continuous map T and a constraint set C with non-empty M_C(X,T) such that the objectives with more than one maximizer form a set with non-empty interior, or a set that is not negligible in the prevalent sense; such a system would directly contradict the generic/prevalent uniqueness theorem.

Watch

Extended reading notes

Core claim

The paper establishes that the constrained optimization problem—maximize ∫φ dμ over T-invariant measures satisfying ∫g dμ = 0 for all g in the constraint set—has a unique solution for a residual and prevalent set of objective functions φ, provided only that the feasible set M_C(X,T) is non-empty. It also proves that non-emptiness is equivalent to the absence of a strictly positive function in the smallest closed T-invariant subspace generated by the constraints, and that optimizers exist whenever the feasible set is non-empty. For constraints of finite type (those definable by finitely many functions), every closed face of the feasible set is realized as the optimizer set of some continuous

Load-bearing premise

The realization theorem depends on an unproved lemma about closed faces of finite-codimensional slices of compact convex sets; if that lemma is false, Theorem 1.9 collapses.

Editorial extensions

If this is right

Whenever M_C(X,T) is non-empty, the objectives with a unique maximizer form both a residual and a prevalent set—uniqueness is the typical outcome, not a rare one.
Non-emptiness of the constrained feasible set has a clean certificate: it fails exactly when the constraint space forces a strictly positive function; otherwise every objective has an optimizer.
For finite-type constraints—covering subsystems, fixed rotation vectors, relative measures, and ergodic optimal transport—every closed face of the feasible set is realized as the optimizer set of some continuous objective.
The dual formula gives a practical bound certificate: any admissible triple (f, g, c) provides an upper bound for the constrained maximum, and the infimum saturates the bound.

Reading between the lines

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

The same generic-uniqueness mechanism may transfer to constrained problems over symbolic systems with locally constant constraints, since the feasible set there is again a slice of the simplex and the correspondence argument should carry over.
The duality formula is an infinite-dimensional linear program; one could test it numerically on finite-type examples by computing the constrained maximum and the infimum over admissible triples.
If the face-intersection lemma used in the realization proof is replaced by a direct proof, the realization theorem would extend beyond finite-type constraints to all compact metrizable systems; the paper's examples all satisfy finite type, so the gap is narrow.
In ergodic optimal transport, the generic uniqueness result suggests that for a typical continuous cost, the optimal invariant coupling between two ergodic measures is unique—a concrete, checkable consequence of the paper's framework.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my quick take for your file. This paper does something genuinely useful: it sets up a single constrained problem — maximize ∫φ dμ subject to ∫g dμ = 0 for g in a subset of C(X) — and shows that unconstrained ergodic optimization, subsystem/rotation-vector problems, relative optimization, and ergodic optimal transport all fit as special cases. The existence criterion (Theorem 1.7) looks right; the duality theorem (Theorem 1.10) is a nice unification; and the centerpiece, Theorem 1.8, generalizes the known generic and prevalent uniqueness results of Jenkinson and Morris to every non-empty feasible set of this linear-constraint type. I read the proof carefully and it appears sound: the upper semicontinuity argument, Fort's theorem for residual continuity, and Christensen's theorem for prevalence all line up. There are minor typos — Theorems 4.4 and 4.7 say 'topological vector space' where they mean 'topological dynamical system', and Lemma 3.2 says 'measure on C(X)' instead of 'on X' — but those are cosmetic.

The real soft spots are two, and both are narrower than they sound. First, Corollary 2.14 claims every example in the introduction has finite type, but the proof leans on the facial property, which only holds for relative ergodic optimization and ergodic optimal transport when the prescribed measure/marginal is ergodic. For non-ergodic ν, the feasible set typically contains no ergodic measures, and finite type generally fails — no finite set of constraints can pin down a non-ergodic marginal. So the realization theorem (1.9) does not cover those cases as advertised. This doesn't touch Theorems 1.7, 1.8, or 1.10, but it does need a correction or a carefully stated hypothesis.

Second, Theorem 1.9 rests on Proposition 5.6, a slice lemma credited to Lazar and cited from Lau without proof. The paper does not verify that the lemma's hypotheses hold for the finite-type slices that arise. The lemma may be true, but the proof of Theorem 1.9 is incomplete as written. A referee should ask the authors to supply a proof or a precise reference with the hypotheses explicitly checked.

Bottom line: the central results are solid and the framework is worth having. The overclaim in Corollary 2.14 and the unproved slice lemma are fixable. I'd send it to a good referee, and I'd expect a revision that tightens the scope of the realization results and supplies the missing lemma. Worth bringing to the group.

Referee Report

3 major / 4 minor

Summary. The paper introduces a constrained version of ergodic optimization. Given a continuous map T on a compact metrizable space X and a constraint set C ⊂ C(X), the feasible set M_C(X,T) consists of T-invariant measures that vanish on C. The main results are: (i) Theorem 1.7, a characterization of nonemptiness of M_C(X,T) and existence of optimizing measures; (ii) Theorem 1.8, asserting that if M_C(X,T) is nonempty, then the set U_C = {φ ∈ C(X) : the constrained maximization has a unique maximizer} is both residual and prevalent in C(X); (iii) Theorem 1.9, a realization theorem stating that for finite-type constraint sets every closed face of M_C(X,T) is the solution set of some continuous objective; and (iv) Theorem 1.10, a duality formula that generalizes Kantorovich duality and known ergodic-optimization duality. The paper positions this framework as a common generalization of unconstrained ergodic optimization, rotations-vector constrained optimization, relative ergodic optimization, and ergodic optimal transport.

Significance. If the main theorems hold, Theorem 1.8 is a valuable generalization of the classical generic/prevalent uniqueness results of Jenkinson and Morris, and it applies to a broad class of constrained problems, including relative ergodic optimization and ergodic optimal transport. The proof of Theorem 1.8 is carefully executed: the upper semicontinuity of the argmax correspondence, the Fort-theorem argument, and the Christensen-based prevalence argument are internally consistent and appear correct. The duality framework (Section 6) is also a useful unification. However, the advertised scope of the realization theorem (Theorem 1.9) is overstated. The claim that Examples 1.4–1.5 always have finite type is not justified and is generally false for non-ergodic prescribed marginals, and the proof of Theorem 1.9 relies on an imported lemma (Proposition 5.6) that is not proved and contains undefined notation. These issues are load-bearing for the realization part of the paper, though they do not affect the soundness of the central uniqueness theorem.

major comments (3)

[Corollary 2.14] The claim that Examples 1.4 and 1.5 have finite type is not supported. Example 2.8 establishes the facial property only for ergodic ν (respectively ergodic μ1, μ2); for non-ergodic marginals the feasible set M_C contains no ergodic measures (Remark 2.7), so the proof via Proposition 2.13 fails. Moreover finite type itself generally fails: if π is the identity and ν has infinitely many ergodic components, no finite set C' can cut out the singleton {ν} in the infinite-dimensional simplex M(X,T). Thus Theorem 1.9 does not apply to the full scope claimed for Examples 1.4–1.5 unless ergodicity assumptions are added or a direct proof of finite type in the non-ergodic cases is supplied.
[Section 5, Proposition 5.6] The realization theorem (Theorem 1.9) depends critically on Proposition 5.6, which is quoted from [33] without proof and contains undefined notation ('H1'). The authors should provide a proof or a precise reference with the exact hypotheses, and verify that M_C(X,T), as a finite-codimensional slice of the simplex M(X,T), satisfies those hypotheses. As written, the proof of Theorem 1.9 cannot be checked from the paper's own arguments.
[Theorem 1.7 / Lemma 3.2] In the proof of Lemma 3.2, the measure μ is described as 'a Borel probability measure on C(X)' but it is used as a measure on X. This appears to be a typo, but it is confusing in a lemma that is used to prove the existence theorem. Please correct to 'on X'.

minor comments (4)

[Theorems 4.4, 4.7, 4.9] The hypotheses state 'Let (X,T) be a topological vector space'; this should be 'topological dynamical system'.
[Proposition 5.6] The undefined symbol 'H1' appears in the statement of Proposition 5.6. It should be replaced with the intended symbol (likely 'M' or the slice's ambient subspace).
[Example 6.4] The derivation of (6.8) from Theorem 6.1 requires replacing φ by −φ. This sign change should be made explicit to avoid confusion.
[Definition 5.3] There is a typographical error: 'the we call' should be 'then we call'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are proved from standard external results and do not reduce to their inputs.

full rationale

The paper's derivation chain is self-contained in the sense relevant to circularity. Theorem 1.7 (existence) is proved by Hahn-Banach separation and a Krylov-Bogolioubov type argument; Theorem 1.8 (generic and prevalent uniqueness) is proved from upper semicontinuity of the argmax correspondence, Fort's theorem, and Christensen's theorem on differentiability of Lipschitz maps; Theorem 1.10 (duality) follows from Kantorovich duality and a minimax theorem. None of these proofs fits a fitted parameter to a subset of data and then 'predicts' a related quantity, and no central object is defined in terms of the conclusion it is used to establish. The only self-citation is [37], which appears in Example 1.4 as contextual motivation ('appeared in the context of statistical inference for dynamical systems') and is not used in any proof, so it is not load-bearing. The realization theorem (Theorem 1.9) relies on Proposition 5.6, an external convex-geometric lemma credited to Lazar via Lau [33] and quoted without proof; if that lemma were false or inapplicable, the proof would collapse, but that is an omitted-verification/correctness concern, not circularity. Similarly, Corollary 2.14's finite-type claim may overstate the scope for Examples 1.4 and 1.5 when the prescribed marginal measure is not ergodic (cf. Example 2.8, which explicitly assumes ergodicity), but that is a mathematical accuracy issue, not a case of the paper's conclusion being equivalent to its assumptions by construction. Overall, no prediction or first-principles result reduces to its own inputs.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no empirical quantities, no new constants, and no new entities. It relies on a standard toolbox of convex analysis and ergodic theory; the only non-elementary external input specific to the field is Lazar's slice lemma. The finite-type condition is a hypothesis, not a fitted parameter.

assumptions (8)

standard math Krylov-Bogolioubov: M(X,T) is non-empty, compact, convex, metrizable; extreme points are ergodic.
Standing structure for the ambient simplex; used in Propositions 2.2, 2.9 and throughout.
standard math Hahn-Banach theorem, including the positive-linear-functional version; Riesz-Markov representation.
Load-bearing in Theorem 1.7 (separation of H and int(P)) and Proposition 6.2 (extension of ν).
standard math Edwards' interpolation theorem for affine functionals on a simplex.
Used in Proposition 2.11 to represent arbitrary compact convex subsets of M(X,T) as constraint feasible sets.
standard math Fort's theorem on residual continuity of semi-continuous correspondences.
Gives residual continuity of the argmax correspondence in Theorem 4.4.
standard math Christensen's theorem: Lipschitz real-valued maps on separable Frechet spaces are Gateaux differentiable off a Haar null set.
Gives prevalence of differentiability in Theorem 4.9.
standard math Adams-Hedberg minmax theorem (Proposition 6.3).
Interchanges inf over measures and sup over H in Theorem 6.1; requires lsc/convex in μ and concave in f.
standard math Davies/Edwards exposed-face theorem: every closed face of a compact metrizable simplex is exposed; Jenkinson's representation of weak* continuous affine functionals by continuous functions.
Bridges convex-geometric exposed faces to continuous objectives in Theorems 1.9 and 5.1.
domain assumption Lazar's slice lemma (Proposition 5.6): closed faces of a finite-codimensional slice are intersections with closed faces of the ambient compact convex set.
Unproved external lemma on which Theorem 1.9's finite-type realization rests; no hypothesis verification is given in the paper.

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Cite this review

Pith. "Pith review of Ergodic Optimization with Linear Constraints." pith.science (2026). https://pith.science/paper/OTXZAFLJ

@misc{pith2026260802435,
  author       = {Pith},
  title        = {Pith review of: Ergodic Optimization with Linear Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OTXZAFLJ}},
  note         = {Machine review of arXiv:2608.02435}
}

abstract

Let $T : X \to X$ be a continuous map of a compact metrizable space, and let $\phi : X \to \mathbb{R}$ be a continuous function. The ergodic optimization problem is to maximize the integral $\int \phi \, d\mu$ as $\mu$ ranges over all $T$-invariant Borel probability measures on $X$. In this paper we consider a constrained version of the ergodic optimization problem. Given a `constraint set' $\mathcal{C}\subset C(X)$, let $M_\mathcal{C}(X,T)$ be the set of $T$-invariant Borel probability measures $\mu$ on $X$ such that $\int g \, d\mu = 0$ for all $g \in \mathcal{C}$. We investigate the problem of maximizing the integral $\int \phi \, d\mu$ over the constrained set $M_\mathcal{C}(X,T)$. We address basic properties of this optimization problem, beginning with nonemptiness of $M_\mathcal{C}(X,T)$ and existence of optimal solutions. Additionally, we establish the generic and prevalent uniqueness of optimal measures, we provide a realization result, and we give a characterization of the dual problem. This framework provides a common generalization of several previously considered optimization problems in dynamical systems and optimal transport.

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Reference graph

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

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2, 193–213

Ka-sing Lau,Infinite dimensional polytopes, Mathematica Scandinavica32(1973), no. 2, 193–213

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314, Springer Science & Business Media, 2012

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