Minimax aspects of optimizations in ergodic theory

Shoya Motonaga

REVIEW 2 major objections 4 minor 8 references

Minimax aspects of optimizations in ergodic theory

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The maximum ergodic average is a finite-time minimax value.

desk verdict New minimax identity for maximum ergodic averages, with proof gaps that are fixable; worth a referee but not a desk reject. read the letter →

arxiv 2411.17615 v2 pith:I3H3WMYB submitted 2024-11-26 math.DS

Shoya Motonaga This is my paper · ORCID

classification math.DS MSC 37A0537A50

keywords minimaxproblemergodicoptimizationmaximumaveragetimeaveragesabstractvariationalprinciplegeneralizedpressurefunctionsFenchel-Rockafellarduality

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 claims that the central quantity of ergodic optimization, the maximum ergodic average, is already determined by finite-time averages before any limit is taken. For a compact dynamical system and continuous potential $\varphi$, it proves $$\$\alpha$(\varphi)=\sup_{x\in X}\inf_{n\in\mathbb{N}}\frac{S_n\varphi(x)}{n}=\inf_{n\in\mathbb{N}}\sup_{x\in X}\frac{S_n\varphi(x)}{n}.$$ The significance is that the exact value can be bracketed by optimizing over finitely many time steps in each direction, and the equality is a genuine minimax identity rather than a limit theorem. The same minimax viewpoint gives a short proof of the abstract variational principle for generalized pressure functions, and the paper shows the underlying minimax theorem is equivalent to the Fenchel-Rockafellar duality in convex analysis.

What carries the argument

The carrying object is the finite-time ergodic average $F(n,\mu)=\int\frac{S_n\varphi}{n}\,d\mu$, viewed as a function on the product of time steps $\mathbb{N}$ and the compact space of Borel probability measures $M(X)$. The minimax theorem for convex/concave functions on such a product supplies the interchange $\inf_n\sup_\mu F=\sup_\mu\inf_n F$; the second characterization $\alpha(\varphi)=\inf_{\psi}\sup_x(\varphi+\psi-\psi\circ T)$ uses the affine space of coboundaries $\psi-\psi\circ T$. For the abstract variational principle, the analogous object is the convex sublevel set $A_\Gamma=\{\xi\in\mathcal{B}:\Gamma(-\xi)\le0\}$ and the entropy-like function $h(\mu)=\inf_{\xi\in A_\Gamma}\int\xi\,d\mu$; the minimax theorem applied to $\int(\xi+\varphi)\,d\mu$ carries the proof.

What would settle it

Compute, for a concrete compact dynamical system and continuous $\varphi$, the three quantities $\alpha(\varphi)$, $\sup_x\inf_n S_n\varphi(x)/n$, and $\inf_n\sup_x S_n\varphi(x)/n$. The theorem asserts they are equal, so any system where the first and second differ, or the first and third differ, refutes it. A targeted search is a point $x$ with $\inf_n S_n\varphi(x)/n$ not attained but $x$ generic for a maximizing measure; if such a point exists, Lemma 3.3 fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that the maximum ergodic average $\alpha(\varphi)=\sup_{\mu\in M_T(X)}\int\varphi\,d\mu$ admits two finite-time minimax characterizations: $\sup_x\inf_n S_n\varphi(x)/n$ and $\inf_n\sup_x S_n\varphi(x)/n$, both equal to $\alpha(\varphi)$. The proof replaces the classical characterization of $\alpha(\varphi)$ as the upper limit over time of the maximal time average with a maxmin/minimax analysis of the function $F(n,\mu)=\int (S_n\varphi/n)\,d\mu$ on $\mathbb{N}\times M(X)$, together with a lemma showing $\sup_x\inf_n S_n\varphi(x)/n=\sup_x\liminf_n S_n\varphi(x)/n$. The same minimax argument, applied to the sublevel set $A_\Gamma=\{\xi: \Gamma(-\xi)\le 0\}$ of a generalized pressure function, reproves the abstract variational principle $\Gamma(\varphi)=\sup_{\mu\in K}(h(\mu)+\int\varphi\,d\mu)$. A final section shows the needed minimax theorem follows from the Fenchel-Rockafellar duality for convex functions.

Load-bearing premise

The proof of Lemma 3.3 assumes that wherever a point's lowest time average is strictly below its long-run lower limit, that lowest value is attained at some finite time $n$; the equality $\alpha(\varphi)=\sup_x\inf_n S_n\varphi(x)/n$ rests on that attainment.

Editorial extensions

If this is right

The maximum ergodic average can be bounded both above and below by finite-time computations: the exact value is $\inf_n\max_x S_n\varphi/n$, so each $n$ gives a rigorous upper bound and each point gives a lower bound.
Ergodic optimization can be organized as a maxmin problem over time and space, without first selecting an invariant measure; this changes the kind of data and algorithms needed.
For generalized pressure functions, the abstract variational principle follows from the minimax theorem under monotonicity, translation invariance, and convexity of $A_\Gamma$; full convexity of $\Gamma$ is not needed, and it emerges from the variational formula.
The coboundary formulation $\alpha(\varphi)=\inf_\psi\sup_x(\varphi+\psi-\psi\circ T)$ identifies the maximum ergodic average as the least uniform constant that bounds $\varphi$ modulo a continuous coboundary.
The minimax theorem used here is exactly the Fenchel-Rockafellar duality in this convex setting, so the ergodic-optimization and pressure results sit inside convex analysis.

Reading between the lines

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

Extension: a numerical scheme could alternate between the two finite-time expressions, using their gap as a certified error bound for approximating $\alpha(\varphi)$.
Testable extension: for non-compact or merely measurable potentials, one can ask whether $\sup_x\inf_n S_n\varphi/n=\alpha(\varphi)$ survives when the infimum of the time averages is not attained; the paper's Example 3.6 shows attainment is the delicate point.
Dual picture: the Fenchel-Rockafellar equivalence suggests viewing invariant measures as primal variables and coboundary corrections as dual variables, connecting ergodic optimization to calibration problems.

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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

Colleague,

Quick take: the paper is worth reading for Theorem 3.2, which gives two finite-time characterizations of the maximum ergodic average α(φ) = sup_x inf_n S_nφ/n = inf_n sup_x S_nφ/n. That identity is genuinely new as far as I know, and it's a nice complement to Jenkinson's limit-based description. The proof strategy is transparent: combine the pointwise ergodic theorem with the elementary minimax inequality. The worked subshift example is helpful, and the Fenchel–Rockafellar duality connection in Section 4 is a good perspective.

The prose is clear and the paper is honest about what is a new proof of a known theorem (Theorem 3.8) rather than a new theorem. Theorem 3.7 is a standard duality formula, and the author doesn't oversell it.

Now the soft spots, in increasing order of seriousness. Lemma 3.3 asserts that when f(x) < g(x) the infimum f(x) is attained at some finite n. That's true, but it isn't automatic: it needs the finite-tail argument — because g(x) is the liminf, the tail averages stay bounded away from f(x), so the infimum is achieved over a finite initial block. The paper omits this, and the proof of Theorem 3.2 depends on the lemma. That's a minor gap, easily fixed.

More substantial is the proof of Theorem 3.8. The author justifies the lower semicontinuity of h̃ by claiming that a pointwise infimum of lower semicontinuous functions is lower semicontinuous — which is false in general. The specific claim happens to be repairable, because h̃ is actually a supremum of affine functionals, hence lsc for the right reason. But as written, the justification is wrong, and a referee should ask for that to be corrected. The minimax application to the variational principle also deserves a careful check of Fan's conditions, though nothing there looks fatal.

Minor: there's a small typo in Example 3.6 where the point a(01)^∞ should read a(10)^∞, and the proof of Proposition 2.1 uses f and F inconsistently.

Overall, the paper is a modest but real contribution. Theorem 3.2 is a clean new characterization, and the abstract variational principle proof is a good idea even if the current write-up is not fully rigorous. I'd send it to a referee; a careful revision should make it publishable.

Recommendation: engage with it, but ask the author to fix the lsc issue and the attainment gap before acceptance.

Referee Report

2 major / 4 minor

Summary. The paper studies minimax formulations of optimization problems in ergodic theory. For a continuous map on a compact metric space and φ ∈ C(X), it characterizes the maximum ergodic average α(φ) as both sup_x inf_n S_nφ(x)/n and inf_n sup_x S_nφ(x)/n (Theorem 3.2), and also as the infimum over coboundary perturbations of the sup norm of φ + ψ − ψ∘T (Theorem 3.7). It then gives a short proof of the abstract variational principle for generalized pressure functions of Bís et al. via Fan's minimax theorem (Theorem 3.8), and describes a connection between this minimax approach and Fenchel-Rockafellar duality (Section 4).

Significance. The main results are parameter-free and the proofs rely only on standard minimax and pointwise ergodic theorems. Theorem 3.2 is a clean, finite-time characterization of the maximum ergodic average, and Theorem 3.7 gives a nice duality-type description. The minimax proof of the known abstract variational principle is conceptually appealing. However, two load-bearing proof points—the attainment step in Lemma 3.3 and the semicontinuity argument in the proof of Theorem 3.8—are not justified correctly as printed. These are repairable, but the manuscript in its present form does not fully support its claims.

major comments (2)

[§3.1, Lemma 3.3 (Eq. (12))] The proof asserts that for x ∈ G with f(x) < g(x), there exists n* ∈ N such that S_{n*}φ(x)/n* = f(x). This is not automatic from the definition of f as an infimum; it requires the finite-tail argument that, because g(x) > f(x), the tail averages are bounded away from f(x), so the infimum is attained in a finite initial segment. This missing justification is load-bearing because the subsequent 'largest n0' construction and the strict comparison after Eq. (12) depend on it. The claim is true, so this is a rigor gap rather than a fatal error, but it must be supplied.
[§3.2, proof of Theorem 3.8] The statement 'Since the pointwise infimum of proper convex and lower semicontinuous functions is also proper convex and lower semicontinuous, so is h̃' is not valid as a general principle, and the operation in this application is actually a pointwise supremum. Indeed, h(μ) = inf_{ξ∈A_Γ} ∫ξ dμ is concave and upper semicontinuous, while h̃(μ) = −h(μ) = sup_{ξ∈A_Γ}(−∫ξ dμ) is convex and lower semicontinuous as a supremum of continuous affine functions. The proof should be corrected to this argument before applying the Fenchel-Moreau theorem.

minor comments (4)

[§2, Theorem 2.2] The statement says 'for every z∈Z, F(x,y) is upper semicontinuous on W'; this should read F(z,w). Also, the remark that 'any cases in [Fan53] cannot be directly applied to Theorem 3.2' is unexplained and should be clarified or removed.
[§3.1, Example 3.6] In the computation of inf_{n∈N} S_nφ(a(10)^∞)/n, the second branch '1/2 + (2a−1)/(2n)' still contains n and is not an evaluation of the infimum; the correct value is a for a ≤ 1/2 and 1/2 for a > 1/2. There is also a typo: 'a(01)' should be 'a(10)'.
[§3.1, Remark 3.4] The sentence 'not all the maximum point of lim inf ... do not take the maximum point of inf ...' contains a double negative; it should read 'not all maximizers of lim inf ... attain the maximum of inf ...'.
[General] There are several typographical and grammatical errors, such as 'miximizing', 'does not coincides with neither ... nor ...', and inconsistent use of 'maximizing' vs. 'maximizing'. A careful language edit is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the minimax characterizations are derived from standard ergodic-optimization results and the minimax inequality, not from their own conclusions.

full rationale

The paper contains no fitted parameters, no normalization choices that force the result, and no reliance on the author's own prior work; the cited results are external (Jenkinson, Fan, Biś–Carvalho–Mendes–Varandas, Rockafellar). Theorem 3.2 is obtained from the established Proposition 3.1, the elementary minimax inequality (Proposition 2.1), and Lemma 3.3, which independently compares inf_n S_nφ(x)/n with liminf_n S_nφ(x)/n via the pointwise ergodic theorem; the target equality is not assumed in that lemma. The proof of the abstract variational principle (Theorem 3.8) is self-contained from axioms (C1)–(C3), the minimax theorem, and Fenchel–Moreau duality, and the required inequality is checked directly rather than imported. Section 4 derives a minimax lemma from Fenchel–Rockafellar duality in a standard way. The one rigor concern, Lemma 3.3's unproved assertion of attainment in Eq. (12), is not circular: the omitted finite-tail argument follows from f(x)<g(x)=liminf and would not use the lemma's conclusion. Thus the derivation chain is self-contained and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities appear. The work relies on standard convex analysis and ergodic theorems, plus the generalized pressure axioms from Biś et al. (2022). The main unstated proof detail is the finite-tail argument in Lemma 3.3.

assumptions (7)

standard math Fan's minimax theorem (Fan 1953): equality of sup-inf and inf-sup for convex-concave functions with one compact Hausdorff factor.
Used in Theorem 3.7 and Theorem 3.8; stated as Theorem 2.2 in Section 2.
standard math Pointwise ergodic theorem for invariant probability measures on compact metric spaces.
Used in Eq. (9) of Theorem 3.2 to obtain a point with liminf equal to the maximum ergodic average.
standard math Existence of an ergodic maximizing measure for φ in M_T(X).
Compactness of M_T(X) and convexity give an extremal maximizer; used in the proof of Theorem 3.2, Eq. (9).
standard math Fenchel-Moreau theorem: proper convex lower semicontinuous functions on a Banach space equal their biconjugate.
Used in Section 3.2 to deduce Eq. (18) and in Section 4 to prove Lemma 4.1.
standard math Banach-Alaoglu theorem: the unit ball of the dual of a normed space is weak-* compact.
Used to ensure K=Ma(X) is compact for the minimax application in Section 3.2.
domain assumption The generalized pressure axioms (C1) monotonicity, (C2) translation invariance, and (C3) convexity on Γ.
Taken from [BCMV22]; they define the class of pressure functions in Section 3.2.
domain assumption The set A_Γ={ξ∈B; Γ(-ξ)≤0} is convex.
Needed for applying Fan's minimax theorem in the proof of Theorem 3.8; the paper notes in Remark 3.9(i) that it follows from (C3), but uses it as a separate hypothesis.

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Cite this review

Pith. "Pith review of Minimax aspects of optimizations in ergodic theory." pith.science (2026). https://pith.science/paper/I3H3WMYB

@misc{pith2026241117615,
  author       = {Pith},
  title        = {Pith review of: Minimax aspects of optimizations in ergodic theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I3H3WMYB}},
  note         = {Machine review of arXiv:2411.17615}
}

read the original abstract

We study optimization problems in ergodic theory from the view point of minimax problems. We give minimax characterizations of maximum ergodic averages involving time averages. Our approach works for the abstract variational principle of generalized pressure functions which is proved by Bi\'{s} et al. (2022). We also describe the relationship between our minimax results and the Fenchel-Rockafellar duality.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

LA CONDITION DE WALTERS

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[2]

Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

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[3]

Rotation, entropy, and equilibrium states

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Every ergodic measure is uniquely maximizing

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Functions for relative maximization

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Ergodic theory on compact spaces

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Ergodic optimization for generic continuous functions

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Zero temperature and selection of maximizing measure, http://www.math.univ-brest.fr/perso/renaud.leplaideur/KATHMANDU.pdf

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

LA CONDITION DE WALTERS

work page

[2] [2]

Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

work page

[3] [3]

Rotation, entropy, and equilibrium states

work page

[4] [4]

Every ergodic measure is uniquely maximizing

work page

[5] [5]

Functions for relative maximization

work page

[6] [6]

Ergodic theory on compact spaces

work page

[7] [7]

Ergodic optimization for generic continuous functions

work page

[8] [8]

Zero temperature and selection of maximizing measure, http://www.math.univ-brest.fr/perso/renaud.leplaideur/KATHMANDU.pdf

work page