{"id":"41163b5f-4251-411d-82e9-d0effaa710cc","arxiv_id":"2411.17615","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximum ergodic average α(φ) equals the supremum over points of the infimum over time of Birkhoff averages, and also equals the infimum over time of the supremum over points.","lead":"The paper shows that the maximum long-run average of a continuous function along an orbit of a dynamical system equals two minimax expressions, one swapping the order of a supremum over points and an infimum over time. It also gives a short minimax proof of a known variational principle for generalized pressure functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.3's attainment step (Eq. 12) is asserted without proof; it is true, so the central theorem survives, but the proof is incomplete as printed.","rationale":"The reader's weakest assumption correctly identifies the unproved attainment step in Lemma 3.3. I checked that the step is true: the gap between f(x) and g(x)=liminf forces all sufficiently late averages to lie strictly above f(x), so the infimum is achieved among finitely many early times. With this one-line argument added, the remainder of Lemma 3.3 is valid: the largest attaining time exists, the strict increase of the induced averages follows from the displayed algebra, the concatenated averages bound g(x) above by β, and the contradiction to sup f<sup g is sound. I also checked the rest of Theorem 3.2. The proof of Eq. (4) is a correct application of the elementary minimax inequality together with the classical limsup estimate (3). The proof of the first identity uses only pointwise ergodic theory and Lemma 3.3; the pointwise ergodic argument does not create circularity because the chain α=liminf S_n(y)/n≤sup_x liminf≤sup_x limsup≤limsup sup≤α uses only (3), which is established independently. The paper contains other minor presentation issues, such as the false blanket statement in Theorem 3.8's proof that pointwise infima of convex lsc functions are convex lsc, but there h̃ is actually a supremum of affine functions, so that proof is also repairable. None of these issues overturns the central minimax characterization, but the printed proof of Lemma 3.3 should be amended. The reader's CONDITIONAL verdict is appropriate; I do not see a basis to move it to ACCEPT or REJECT.","tokens_in":8284,"tokens_out":28479,"duration_ms":255158,"concrete_test":"Re-derive Eq. (12) formally: fix x with f(x)<g(x), choose ε=(g(x)-f(x))/2 and N such that S_nφ(x)/n ≥ f(x)+ε for all n≥N; then f(x)=min_{1≤n≤N} S_nφ(x)/n, which is attained. Then replay Lemma 3.3 using this finite-tail min. If the min were not attained, exhibit n_j→∞ with S_{n_j}φ(x)/n_j→f(x) and observe g(x)≤f(x), contradicting f(x)<g(x); this verifies that the n_i construction in Lemma 3.3 is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.2 rests on Lemma 3.3, and the construction in Lemma 3.3 begins with the assertion in Eq. (12) that for x with f(x)<g(x), the infimum f(x)=inf_{n∈N} S_nφ(x)/n is attained at some n*. This is not automatic from the definition alone, and the paper gives no argument. The claim is true: since g(x)=liminf_{n→∞} S_nφ(x)/n > f(x), there is ε>0 and N such that S_nφ(x)/n ≥ f(x)+ε for every n≥N. Hence the infimum is the minimum over the finite set 1≤n≤N and is attained. If this finite-tail argument were false, a sequence n_j→∞ with S_{n_j}φ(x)/n_j→f(x) would force g(x)≤f(x), contradicting f(x)<g(x). Thus the set of attaining times is finite and nonempty, so the largest n0 exists and the strict-comparison step after Eq. (12) is valid. The concern is therefore about rigor rather than validity: with the missing justification supplied, the orbit-chain argument and the contradiction to sup f<sup g go through, and Theorem 3.2 stands.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":8525,"tokens_out":17958,"duration_ms":144671,"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":[{"comment":"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.","section":"§3.1, Lemma 3.3 (Eq. (12))"},{"comment":"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.","section":"§3.2, proof of Theorem 3.8"}],"minor_comments":[{"comment":"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.","section":"§2, Theorem 2.2"},{"comment":"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)'.","section":"§3.1, Example 3.6"},{"comment":"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 ...'.","section":"§3.1, Remark 3.4"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper's contribution is modest but potentially useful. The two proof gaps identified are fixable and do not appear to threaten the validity of the main theorems. With those corrections and a careful language edit, the paper would be acceptable for publication. My only additional concern is that the novelty over the known results of Bís et al. and Jenkinson is mainly expository, but the minimax characterizations and the Fenchel-Rockafellar connection do add a fresh perspective."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick 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.\n\nThe 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.\n\nNow 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.\n\nMore 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.\n\nMinor: 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.\n\nOverall, 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.\n\nRecommendation: engage with it, but ask the author to fix the lsc issue and the attainment gap before acceptance.","headline":"New minimax identity for maximum ergodic averages, with proof gaps that are fixable; worth a referee but not a desk reject.","tokens_in":9068,"tokens_out":4196,"would_cite":true,"duration_ms":35734,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A05","37A50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The maximum ergodic average is a finite-time minimax value.","keywords":["minimax problem","ergodic optimization","maximum ergodic average","time averages","abstract variational principle","generalized pressure functions","Fenchel-Rockafellar duality"],"falsifier":"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.","tokens_in":8068,"feed_emoji":"🎲","tokens_out":9433,"duration_ms":95141,"temperature":0.7,"pith_summary":"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.","feed_headline":"Maximum ergodic average is a finite-time minimax value","feed_subtitle":"Two finite-time formulas replace limits in ergodic optimization and prove the pressure variational principle.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the classical characterization of the maximum ergodic average as the upper limit of maximal time averages (Proposition 3.1), which Theorem 3.2 improves.","marker":"[Jen06]"},{"why":"Provides the minimax theorem used to interchange the supremum over measures with the infimum over time steps.","marker":"[Fan53]"},{"why":"Defines generalized pressure functions and states the abstract variational principle that Section 3.2 reproves via the minimax approach.","marker":"[BCMV22]"},{"why":"Correction and companion to [BCMV22], cited alongside it for the abstract variational principle.","marker":"[BCM+23]"},{"why":"Gives the Fenchel-Rockafellar duality used to prove Lemma 4.1 and hence the minimax equalities.","marker":"[Roc66]"}],"fun_headline_variants":["Maximum ergodic average equals finite-time minimax value","Two finite-time minimax formulas for ergodic optimization","Finite-time minimax proves pressure variational principle","Minimax duality yields finite-time ergodic optimization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Maximum ergodic average equals finite-time minimax value","Two finite-time minimax formulas for ergodic optimization","Finite-time minimax proves pressure variational principle","Minimax duality yields finite-time ergodic optimization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000996,"raw_usage":{"total_tokens":4162,"prompt_tokens":834,"completion_tokens":3328,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":3266}},"tokens_in":450,"tokens_out":3328,"duration_ms":19405,"temperature":1.0,"reasoning_tokens":3266,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:58:49.483262+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}