{"id":"04d015b5-2454-4f59-82fd-65776cb9b55e","arxiv_id":"1908.08270","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every weakly coarse expanding system, every Holder potential has a unique equilibrium state and the system satisfies the central limit theorem, the law of the iterated logarithm, exponential decay of correlations, and large deviations, including on the 2-sphere with periodic critical points.","lead":"Weakly coarse expanding systems, a wide class that includes expanding Thurston maps and many non-hyperbolic rational maps, are shown to admit a unique equilibrium state for every Holder potential, together with a central limit theorem, law of the iterated logarithm, exponential decay of correlations, and large deviations. The same results hold on the 2-sphere even when periodic critical points are present, via a blow-up construction.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The large-deviation formula in Theorem 1.1(5) is ill-posed and does not match the standard shift LDP; claim (5) is not correct as stated.","rationale":"The reader identified strong path connectivity as the weakest assumption, but that is an explicit hypothesis in the definition of a weakly coarse expanding system and does not threaten the theorem as stated. My concern is different and more direct: one of the seven headline assertions, the large deviation principle in Theorem 1.1(5), is displayed with a function-valued term `tψ` on both sides, so it is not a real-valued identity. The proof of the statistical laws is a transfer from the full shift, so the formula should reduce to the classical shift LDP. Testing the displayed expression against the exactly solvable full-shift case shows a numerical mismatch for any natural correction. This means the theorem statement, as written, asserts a false or ill-posed LD claim, even though the rest of the argument—unique equilibrium states, CLT, LIL, EDC—may well be sound. The appropriate disposition is to require the authors to correct Theorem 1.1(5) and supply the valid LDP (probably the standard Legendre-transform rate) before acceptance. I also noted a smaller proof-level typo in Proposition 2.16 (the base point appears to be w_{i_1} where w is needed), but that is easily repaired and is not the basis of my objection.","tokens_in":30892,"tokens_out":49717,"duration_ms":513804,"concrete_test":"Analytical check: take the one-sided full shift on {0,1}, φ≡0, ψ=1_{first coordinate}, and compute both sides of Theorem 1.1(5) for t=0.1. The exact LDP by Cramér gives lim_{n→∞} (1/n) log P(S_nψ ≥ 0.6n) = −[0.6 log(1.2)+0.4 log(0.8)] ≈ −0.0202, while plausible corrections of the displayed RHS give +0.0012 or +0.1012; the displayed event also fails to be numeric because of `tψ`. If the authors provide a corrected formula, verify it reproduces the Cramér rate for this example and that the transfer proof in Section 4 establishes that formula.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.1(5) is part of the central claim, but as printed it is not a well-formed statement: both the event and the right-hand side contain the term `tψ(x)`, so the left side is a number while the right side, if meaningful at all, depends on the point x. Moreover, the transfer argument in Section 4 derives all statistical laws from the full shift, so the displayed LD rate must coincide with the classical Cramér rate for the full shift. It does not: for the full shift on {0,1}, φ≡0, ψ=1_{first coordinate}, and t=0.1, the exact rate for {S_nψ ≥ 0.6n} is −D(0.6||0.5) ≈ −0.0202. Under the minimal correction `tψ→0`, the displayed RHS is P(φ+0.1ψ)−P(φ)−0.1∫ψ = log(1+e^{0.1})−log2−0.05 ≈ +0.0012; under `tψ→t` it is ≈ +0.1012. Neither equals the Cramér rate. Thus the paper's LD statement is not the LDP that the proof can inherit from the shift, and it needs correction or replacement. This is a headline-claim defect, not merely a typo in the proof.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops thermodynamic formalism for weakly coarse expanding systems: continuous finite branched covers of locally compact strongly path connected spaces satisfying topological expansion and irreducibility. Its main results assert existence and uniqueness of equilibrium states for Hölder potentials and the validity of the central limit theorem, law of the iterated logarithm, exponential decay of correlations, and a large deviation principle, both for systems without periodic critical points (Theorem 1.1) and for open subsets of S² with periodic critical points allowed (Theorem 1.2). The strategy is to encode the system by a geometric coding tree semiconjugate to the full shift, prove that entropy does not drop under the coding map, and transfer statistical laws from the shift; in the S² case, periodic critical points and their grand orbits are blown up to circles, producing a weakly coarse expanding system without periodic critical points.","tokens_in":31193,"tokens_out":13422,"duration_ms":137957,"significance":"The framework unifies and extends earlier work on expanding Thurston maps and coarse expanding conformal systems, allowing postcritically infinite behavior and non-conformal dynamics. The paper contains detailed constructive proofs, including two independent constructions of an exponentially contracting metric (via Frink's lemma and an explicit chain metric), and the coding-tree/no-entropy-drop argument is robust. If the statistical laws are stated correctly, the paper would be a valuable contribution to the thermodynamic formalism of non-uniformly hyperbolic systems. The present version, however, contains a defective large-deviation statement in the headline theorem, so the paper needs revision.","major_comments":[{"comment":"The displayed large-deviation assertion in Theorem 1.1(5) is not a well-formed statement. The event contains the expression tψ(x) on the right-hand side of the inequality, and the right-hand side of the limit also contains tψ, so the claimed limit is point-dependent and the equality is not meaningful as a statement about a fixed real t. Moreover, even under the minimal correction tψ→0 on the right-hand side, the formula does not give the rate inherited from the full shift. For the full shift on two symbols, φ≡0, ψ=1_{first coordinate}, and t=0.1, the exact rate for {S_nψ ≥ 0.6n} is −D(0.6‖0.5) ≈ −0.0202, whereas the displayed right-hand side evaluates to P(0.1ψ)−P(0)−0.1∫ψ ≈ +0.0012 under the first correction and to ≈ +0.1012 if tψ is replaced by t. Neither equals the Cramér rate, and the sign is wrong in the first interpretation. The transfer argument in Section 4 can at best yield the standard LDP for the full shift, so Theorem 1.1(5) should be replaced by a correct statement, for example with the rate given by the Legendre transform associated to t↦P(φ+tψ)−P(φ)−t∫ψ. This is a load-bearing defect in the main theorem, not a typographical issue in a proof.","section":"Theorem 1.1(5); Section 4, proof of Theorem 1.1(1)-(5)"},{"comment":"Theorem 1.2 asserts that claim (5) of Theorem 1.1 holds also in the presence of periodic critical points, and the proof in Section 5.1 says that statistical laws follow from the shift by pushforward. Since claim (5) as printed is ill-posed and the displayed rate is not the shift rate, this proof cannot establish the stated theorem. The correction to Theorem 1.1(5) must be propagated to Theorem 1.2; otherwise the main theorems remain defective even if the rest of the thermodynamic formalism argument is sound.","section":"Section 5.1, proof of Theorem 1.2(1)-(5)"}],"minor_comments":[{"comment":"In the proof of Lemma 4.3, the equality \"π⋆~µ = ~µ\" should read \"π⋆~µ = µ\"; the current text appears to be a typo.","section":"Section 4, Lemma 4.3"},{"comment":"The symbol ~W0 is used both for the blown-up space and for the lifted cover (\"Denote this cover by ~W0\"), which creates ambiguity in Lemma 5.3; a different letter, such as ~U0, should be used for the cover.","section":"Section 5, after Proposition 5.1"},{"comment":"The function sgn(t) is undefined at t=0; the corrected large-deviation statement should specify t∈R\\{0} or include a separate convention for t=0.","section":"Theorem 1.1(5)"},{"comment":"The phrase \"a curve which is the branch of f^{-(n-1)}(γ_{i_n})\" is not formally precise, since f^{-(n-1)} of a curve is not a single curve; it should say that γ_n(α) is the lift of γ_{i_n} by f^{n-1} starting at z_{n-1}(α).","section":"Section 2.6, proof of Proposition 2.16"},{"comment":"The parameter c is introduced as \"a constant which will be determined later\" but the proof only chooses it in Step 3; the text should state explicitly that c can be made small enough to satisfy both cd²e^{α₂}<1 and cd²<1.","section":"Appendix A, after equation (14)"}],"recommendation":"major_revision","confidential_remarks":"The paper is in good shape aside from the large-deviation statement, which is part of the headline theorem. I would not reject on this basis: the coding argument can supply a correct LDP from the full shift, so the fix should be local. However, because the displayed statement is not a well-formed theorem and the rate is wrong, major revision is appropriate rather than minor revision or acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this is a substantial paper and the main results are mostly believable. The new class of weakly coarse expanding systems is a useful generalization, and the geometric coding tree plus the blow-up of periodic critical points is a genuine technical achievement. The no-entropy-drop lemma (Lemma 3.2) is clean, and the two independent constructions of an exponentially contracting metric (Frink's lemma and the explicit metric in Appendix A) show real care. I found no circularity: the shift-space facts cited to [24] and Bowen are standard and independently verifiable, and the self-citations are appropriate.\n\nThe trouble is Theorem 1.1(5), the large-deviation claim. As printed, the event contains a free 'tψ' on the right side and the displayed rate also contains 'tψ', so the claimed limit depends on x rather than being a number. That is ill-posed. Worse, even a naive guess that 'tψ' should be 't' does not fix the mathematics: the expression -t∫ψ dµ + P(φ+tψ) - P(φ) is not the level-1 rate function for the empirical average under an equilibrium state; for a full shift with φ≡0 and ψ the first-coordinate indicator, the displayed value has the wrong sign and magnitude. The proof in Section 4 only says the sequence satisfies the LDP by transfer from the shift; it never derives the displayed rate. So the theorem needs a genuine correction or replacement, not a one-character typo fix. I suspect this is repairable, but it is a real defect in a headline claim.\n\nThe rest of the paper holds up well. The blow-up construction in Section 5 is intricate and I did not independently verify every topological claim, but the overall architecture is coherent and the exposition is honest about hypotheses. The strong path connectedness assumption (Definition 2.3) is restrictive and rules out simple examples like trees, but it is explicitly flagged and is a reasonable price for the coding tree.\n\nThis paper deserves a serious referee. I would send it to peer review rather than desk reject, but the referee should insist that Theorem 1.1(5) be fixed or removed, and that the proof provide the actual rate function. After that, I would expect the paper to be accepted.","headline":"Genuinely new and mostly sound paper on equilibrium states for weakly coarse expanding systems, but Theorem 1.1(5) as printed is an ill-posed large-deviation statement that needs to be corrected.","tokens_in":31719,"tokens_out":5800,"would_cite":true,"duration_ms":54384,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D35","37F10","37A50","37B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Weakly coarse expanding dynamical systems admit unique equilibrium states for every Hölder potential, along with the central limit theorem, law of iterated logarithm, exponential decay of correlations, and large deviations.","keywords":["thermodynamic formalism","equilibrium states","coarse expanding dynamical systems","postcritically infinite","central limit theorem","law of iterated logarithm","large deviations","geometric coding tree"],"falsifier":"Find (or prove impossible) a weakly coarse expanding system on a compact, locally connected space that is path connected but not strongly path connected—for instance, a finite tree-like branched cover—and test whether a Hölder potential on its repellor has a unique equilibrium state. If such a system admits two equilibrium states, strong path connectivity is a necessary assumption, not an artifact of the proof.","tokens_in":30711,"feed_emoji":"🌀","tokens_out":10984,"duration_ms":104494,"temperature":0.7,"pith_summary":"The paper brings thermodynamic formalism to a broad class of expanding, possibly postcritically infinite maps: it proves that every Hölder continuous potential on the repellor has a unique equilibrium state. The class of weakly coarse expanding systems includes previously studied expanding sphere maps and coarse expanding conformal systems, but drops finiteness of the postcritical set and does not assume conformality, holomorphy, or smoothness. The proof encodes the dynamics by a geometric coding tree that semiconjugates the system to a full shift, transfers equilibrium states and statistical laws from the shift, and then handles periodic branch points by blowing them up to circles. If the theorems are correct, any Hölder observable on such a repellor is statistically tame: Gaussian fluctuations, iterated-logarithm growth, exponential decay of correlations, and large deviations all follow automatically.","feed_headline":"Unique equilibrium states proven for coarse expanding maps","feed_subtitle":"Even with infinite postcritical sets, a coding tree yields CLT, LIL, and decay of correlations.","key_machinery":"The argument runs through two constructions. First, paths in $W_0$ joining a basepoint to each of its $d$ preimages and avoiding the countable post-branch set build a geometric coding tree: the branches give a Hölder semiconjugacy $\\pi$ from the full shift $\\Sigma$ onto the repellor $X$, and a no-entropy-drop lemma (the number of depth-$n$ cylinders over a point grows subexponentially when no critical point is periodic) lets equilibrium states and statistical laws be pulled back from the shift. Second, for periodic critical points on the sphere, each such point and its grand orbit is blown up to a circle, giving a system without periodic critical points on a carpet-like space; a metrization lemma produces the exponentially contracting metric on the new repellor.","core_discovery":"The paper introduces weakly coarse expanding dynamical systems—continuous finite branched coverings of locally connected, strongly path connected spaces that expand in a weak metric sense—and proves that whenever no critical point is periodic, every Hölder continuous potential $\\phi$ on the repellor $X$ has exactly one equilibrium state, and every Hölder observable $\\psi$ satisfies the central limit theorem, the law of the iterated logarithm, exponential decay of correlations, and the large deviation principle. If $W_0$ is an open subset of the 2-sphere, the same conclusions hold even with periodic (repelling) branch points: the authors blow up each periodic critical point and its grand orbit to circles, obtaining a carpet-like space with no periodic critical points, then run the non-critical argument.","pith_inferences":["This suggests that strong path connectivity could be relaxed to the existence of a finite covering by path connected sets whose intersections remain path connected after removing countable sets; if a coding tree could be replaced by such a covering, the theorems would extend to dendrites and trees.","The paper's criterion for zero variance in the CLT is immediately testable numerically on a postcritically infinite rational map: compute sample variances of a Hölder observable along orbits and check whether they vanish precisely for coboundaries.","The blow-up construction points to a two-way dictionary between periodic branch points and invariant circles, which could generate new examples by starting from a circle map on a carpet and collapsing the invariant circles.","Since the proofs use only oscillation decay along pullback covers, the results should extend to observables with weaker moduli of continuity than Hölder, as long as the topologically Hölder condition is met."],"forward_implications":["Every Hölder potential on the repellor has a unique equilibrium state, and it is the pushforward of the unique Gibbs state for a Hölder potential on the full shift.","Every Hölder observable obeys the central limit theorem, law of iterated logarithm, exponential decay of correlations, and a large deviation principle with rate function given by topological pressure.","The variance in the CLT is zero exactly when the centered observable is a continuous coboundary, and two potentials share an equilibrium state exactly when they differ by a coboundary plus a constant.","All of this remains true on an open subset of the 2-sphere even when periodic branch points are present.","The results apply to a class of examples—iterated function systems, semigroup skew products, maps on carpet-like fractals, expanding polynomial-like maps—that need not be conformal, holomorphic, or postcritically finite."],"supporting_citations":[{"why":"Defines the expansion and irreducibility axioms and the visual metric framework that the paper's weakly coarse expanding class generalizes.","marker":"[15]"},{"why":"Provides the expanding sphere-map setting and the construction of visual metrics that motivate the new class.","marker":"[4]"},{"why":"Supplies the geometric coding tree idea used to construct the semiconjugacy from the full shift to the repellor.","marker":"[22]"},{"why":"Gives the thermodynamic formalism for the full shift, including equilibrium uniqueness, CLT, LIL, exponential decay of correlations, and the metrization lemma.","marker":"[24]"},{"why":"Establishes uniqueness of equilibrium states and the Gibbs formalism on the shift used in the main theorems and the cohomological equation.","marker":"[5]"},{"why":"Provides the metrization lemma used to produce exponentially contracting metrics on the repellor and on the blown-up carpet.","marker":"[13]"}],"fun_headline_variants":["Equilibrium states proven for weakly coarse expanding maps","Infinite postcritical sets tamed: equilibrium states and CLT","Statistical laws for coarse expanding maps with infinite postcritical sets","Coarse expanding dynamics: thermodynamic formalism without finiteness","New results for expanding maps: unique equilibrium states and CLT"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire construction rests on the phase space being strongly path connected: after deleting any countable set of points, the space must remain path connected, because the coding tree is built from paths that avoid the countable post-branch set. A merely path connected space, such as a tree, can be disconnected by removing a single point.","fun_headline_variants_meta":{"raw":{"variants":["Equilibrium states proven for weakly coarse expanding maps","Infinite postcritical sets tamed: equilibrium states and CLT","Statistical laws for coarse expanding maps with infinite postcritical sets","Coarse expanding dynamics: thermodynamic formalism without finiteness","New results for expanding maps: unique equilibrium states and CLT"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000694,"raw_usage":{"total_tokens":3064,"prompt_tokens":792,"completion_tokens":2272,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":408,"completion_tokens_details":{"reasoning_tokens":2191}},"tokens_in":408,"tokens_out":2272,"duration_ms":17728,"temperature":1.0,"reasoning_tokens":2191,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:44:53.161591+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find (or prove impossible) a weakly coarse expanding system on a compact, locally connected space that is path connected but not strongly path connected—for instance, a finite tree-like branched cover—and test whether a Hölder potential on its repellor has a unique equilibrium state. If such a system admits two equilibrium states, strong path connectivity is a necessary assumption, not an artifact of the proof.","supporting_citations":[{"cited_title":"Ha ¨ ıssinsky, K","cited_arxiv_id":null,"evidence_quote":"Defines the expansion and irreducibility axioms and the visual metric framework that the paper's weakly coarse expanding class generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the expanding sphere-map setting and the construction of visual metrics that motivate the new class."},{"cited_title":"Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent","cited_arxiv_id":null,"evidence_quote":"Supplies the geometric coding tree idea used to construct the semiconjugacy from the full shift to the repellor."},{"cited_title":"Przytycki, M","cited_arxiv_id":null,"evidence_quote":"Gives the thermodynamic formalism for the full shift, including equilibrium uniqueness, CLT, LIL, exponential decay of correlations, and the metrization lemma."},{"cited_title":"Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomor phisms, Lecture Notes in Mathematics 470, Springer, 1975","cited_arxiv_id":null,"evidence_quote":"Establishes uniqueness of equilibrium states and the Gibbs formalism on the shift used in the main theorems and the cohomological equation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the metrization lemma used to produce exponentially contracting metrics on the repellor and on the blown-up carpet."}],"review_version":1}