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REVIEW 4 major objections 5 minor 38 references

Equilibrium Stability for Open Zooming Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Equilibrium states of open zooming systems survive limits of their defining data, and are unique.

desk verdict The advertised open-system stability result is not proven: Theorem A's proof never mentions the hole, and the key pressure semicontinuity lemma is a garbled derivation. read the letter →

arxiv 2502.08693 v2 pith:FQKUHECI submitted 2025-02-12 math.DS

Rafael A. Bilbao , Eduardo Santana This is my paper · ORCID
classification math.DS MSC 37D2537D3537B99
keywords equilibriumstatesstabilityopensystemszoomingnon-uniformexpansionskew-productsVianamapsthermodynamicformalism
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 tries to prove that equilibrium states, the invariant probability measures that maximize entropy plus potential, do not jump when the dynamics and the potential are perturbed continuously. For a broad family of open zooming systems, including non-uniformly expanding maps with critical sets and holes, any limit of equilibrium states is again an equilibrium state of the limiting system (Theorem A), and the same holds for fiber-contracting skew-products built over them (Theorem B). A direct consequence, combined with known finiteness of equilibrium states, is uniqueness: each admissible potential has exactly one equilibrium state (Theorem C). The result matters because equilibrium stability is what makes thermodynamic quantities computable and robust under approximation, for example in Viana maps and open systems with holes.

What carries the argument

The central object is a zooming system: a measurable map $f:M\to M$ with a reference zooming measure $\mu$ such that $\mu$-almost every point has infinitely many zooming times, i.e. times at which $f^n$ maps a pre-ball homeomorphically onto a ball of fixed radius while backward iterates contract distances according to a zooming contraction $(\alpha_n)$. The proof machinery is a pressure-semicontinuity argument: a generating finite partition $\mathcal P$ with atoms of diameter below the zooming scale and $\mu_0(\partial\mathcal P)=0$ gives the upper entropy bound $\limsup_n h_{\mu_n}(f_n)\le h_{\mu_0}(f,\mathcal P)$, while a lower bound on the limit pressure comes from local entropy via Brin–Katok and from the $C^0$ convergence of dynamical balls. Together with potentials shifted by constants so that their pressures vanish, these estimates identify every accumulation point as an equilibrium state. For skew-products, the load-bearing identity is the Ledrappier–Walters formula $h_{\tilde\mu}(F)=h_\mu(f)$ for measures projecting to $\mu$ when the fiber map is uniformly contracting.

What would settle it

Compute the open pressure and equilibrium states explicitly for the one-sided shift with the polynomial contraction of Section 6.6, using holes that intersect the zooming set at each finite stage but not in the limit. If the weak-$*$ limit of the finite-Markov-structure equilibrium states assigns positive mass to the limiting hole, or if $\limsup_n P_{f_n,H_n}(\varphi_n) < P_{f,H}(\varphi)$, then the open-system stability statement fails, even though the closed-pressure part of the proof could still hold.

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Extended reading notes

Core claim

The core claim is Theorem A: the family $\mathcal{F}_Z = \{(f,\varphi)\mid f,\varphi \text{ both zooming and }\varphi \text{ has locally Hölder induced potential}\}$ is equilibrium stable. In detail, whenever $(f_n,\varphi_n)\in \mathcal{F}_Z$ converges in the $C^0$ topology to $(f,\varphi)$ and $\mu_n$ is an equilibrium state for each $n$, every weak-$*$ accumulation point $\mu_0$ of the measures $\mu_n$ is an equilibrium state for $(f,\varphi)$. The proof normalizes each potential by subtracting the pressure $P_{f_n}(\varphi_n)$, obtains an entropy upper bound from a generating partition whose atoms shrink on the zooming set, and establishes a lower semicontinuity inequality $\limsup_n P_{f_n}(\varphi_n) \ge P_f(\varphi)$ through Brin–Katok local entropy estimates; the two inequalities then force $\mu_0$ to attain the limit pressure. Theorem B transfers the result to skew-products by projecting invariant measures onto the base and using the Ledrappier–Walters formula, and Theorem C upgrades finiteness plus stability to uniqueness by perturbing each potential so that it selects a single equilibrium state.

Load-bearing premise

The load-bearing premise is that a compatible hole can be chosen uniformly across a $C^0$-convergent sequence so that open equilibrium states stay supported off the hole and converge to an open equilibrium state of the limit; the paper states the family without proving such hole stability.

Editorial extensions

If this is right

  • Equilibrium states in $\mathcal{F}_Z$ vary continuously with $(f,\varphi)$: small $C^0$ changes in the map or potential move the equilibrium states by a small amount in the weak-$*$ topology.
  • Every potential in the family has a unique equilibrium state, because finiteness from the cited equilibrium-state theorem plus stability yields uniqueness through Theorem C.
  • Viana maps, which are non-uniformly expanding maps with critical sets, therefore have a unique measure of maximal entropy; prior results gave only countably many ergodic measures of maximal entropy.
  • The stability passes to fiber-contracting skew-products over zooming bases, so their equilibrium states are continuous and unique as well.
  • The conclusions cover the examples in Section 6: Benedicks–Carleson maps, Rovella maps, local diffeomorphisms with non-uniform expansion, and shifts on metric spaces with polynomial contraction.

Reading between the lines

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

  • The proof establishes the closed-pressure statement and the paper states the result for open systems; a complete open-system formulation would make explicit a uniform choice of holes so that open equilibrium states pass to the limit.
  • The same pressure-semicontinuity mechanism should extend to non-autonomous or random sequences of zooming maps, because the key estimates are uniform in the $C^0$ topology rather than specific to deterministic iteration.
  • Replacing local Hölder regularity of the induced potential by a weaker summability condition, such as an $\ell^1$ variation bound, is a natural testable weakening; if the entropy estimates survive, Theorem A would cover larger potential classes.
  • The fiber-contraction hypothesis in the skew-product theorem could be relaxed to allow fibers with positive entropy as long as the relative entropy vanishes in the limit, since the Ledrappier–Walters formula would still carry the argument.
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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

4 major / 5 minor

Summary. The paper studies equilibrium stability for open zooming systems. It defines open pressure and open equilibrium states for a map with a hole H (Definition 2.3.1), introduces a family F_Z of pairs (f,φ) in which both the map and the potential are zooming and the induced potential is locally Hölder (Section 2.4), and claims in Theorem A that F_Z is equilibrium stable. Theorem B claims an analogous stability result for skew-products whose base is a zooming system, and Theorem C derives uniqueness of equilibrium states from finiteness plus stability. The proofs follow the strategy of Alves--Ramos--Siqueira [4] and Araujo [8]: prove invariance of accumulation points, establish semicontinuity of pressure via a generating partition and a Brin--Katok-type argument, and then compare pressures. The paper also lists several classes of examples (Viana maps, Benedicks--Carleson maps, Rovella maps, shifts with zooming metrics, uniformly expanding maps).

Significance. If the main results were correct, they would extend the existing theory of equilibrium stability from uniformly or non-uniformly expanding closed systems to open zooming systems with special holes, and to skew-products over such bases. The paper has the merit of building on the prior finiteness result [34] and of using a common framework (zooming times, special holes, locally Hölder induced potentials) that covers a wide class of examples. However, the central theorem as written is not actually proved: the proof of Theorem A never uses the hole or the open pressure, and the key semicontinuity lemma contains an invalid derivation. These are load-bearing gaps, not presentation issues. The paper also reproduces examples from earlier work rather than providing new applications that test the open-system statement.

major comments (4)
  1. [§2.4 and §3 (Theorem A)] Theorem A is stated for the family F_Z = {(f,φ): f,φ are both zooming and φ has locally Hölder induced potential}, which contains no hole H, whereas the open pressure P_{f,H} and open equilibrium states in Definition 2.3.1 depend on a hole H and on the class M_f(M,H) = {η: η(H)=0}. In the proof, the pressure is introduced as P_{f_n}(φ_n) := sup_{η∈Z_n}{h_η(f_n)+∫φ_n dη}, i.e. the closed zooming pressure over zooming measures, and the hole H never appears. Consequently the argument proves at most stability of closed equilibrium states; it does not show that accumulation points of open equilibrium states avoid the limiting hole. Theorem 2.3.1 gives open equilibrium states only under the additional hypothesis that the zooming set Λ is not dense and the hole is chosen disjoint from Λ, a hypothesis not included in F_Z and not checked along the convergent sequence (f_n,φ_n). The statement and proof need a precise description of the holes H_n and a proof that the associated open equilibrium states and pressures pass to the limit.
  2. [§3, Lemma 3.0.3] The derivation of the semicontinuity inequality lim sup_{n→∞} P_{f_n}(φ_n) ≥ P_f(φ) is not a valid chain of substitutions. The proof introduces parameters α<0<β<1, writes an expression containing (α sin²β) and P_{f_n}(-φ_n/2(1-cos(β-ε))), then sets α=-1 and lets β→0 inside the limsup without any argument justifying interchange of limits. The intermediate inequalities are also formally inconsistent: after bounding P_f(φ) ≤ limsup h_{ν_n}(f_n), the next line adds terms depending on β and P_{f_n} with signs that do not follow from the previous inequality. Since this inequality is the key pressure comparison on which the proof of Theorem A rests, the proof of Theorem A is incomplete.
  3. [§5, Lemma 5.0.1] The proof of finiteness of the set of all equilibrium states contains unsupported topological claims: it asserts that the set of equilibrium states is 'closed with countable boundary' and that a 'countable perfect set' is finite. A nonempty perfect subset of a compact metric space is uncountable, so if the boundary is countable and perfect it must be empty; in any case the inference to finiteness is not justified by the statements given. The subsequent argument that all continuous zooming potentials have equilibrium states among a finite list relies on this step. This does not affect Theorem C's conclusion directly if Theorem A were established, but as written the proof is incomplete.
  4. [§2.5 and §4 (Theorem B)] Theorem B is stated for the family S that includes both case i.) (uniform contraction on fibres) and case ii.) (plus a common fixed point y0 with g(x,y0)=y0 for all x). Proposition 4.0.1(1) constructs u using a fixed point y0 and then uses g(x,y0)=y0, which is exactly condition ii.). For case i.) no such common y0 is assumed, so the induced potential ~φ on the base is not defined by the given formula. The proof later restricts to potentials constant on the fibres for case i.), but Theorem B claims stability for all zooming potentials in (F_Z)'. The discrepancy between the stated scope and the proof needs to be resolved.
minor comments (5)
  1. [Throughout] The notation for the limit map is inconsistent: the proof of Theorem A writes f0 in a few places (e.g., 'equilibrium state for the system (f0,φ0)') but otherwise uses f; please unify the notation.
  2. [Definition 2.3.3] The induced potential is defined as φ(x)=Σ_{j=0}^{R(x)-1} φ(f^j(x)); the left-hand side should be ar φ(x) (or another symbol) to avoid conflicting with the original potential φ.
  3. [Lemma 3.0.2] The lemma states µ(B_f(x,n,ε))=η_p(B_f(x,n,ε)), but µ has not been introduced; presumably it should be η_{k_j} or a subsequential limit. Also, the sentence about 'a sum of uncountably many positive numbers' should be replaced by a correct continuity-set argument.
  4. [Sections 6.5 and 6.6] The text says 'we apply our Theorem A to obtain an open zooming system and a Markov structure adapted to a hole'; however Theorem A is a stability statement, not a construction result. The existence of Markov structures is Theorem 2.2.1 (from [34]), so the attribution should be corrected.
  5. [Throughout] There are numerous typographical errors (e.g., 'dynamamical', 'equilibrim', 'Raf Ael A. Bilbao' in the header, incomplete parentheses in the introduction's description of the escape rate). A careful proofreading is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem A's proof is a self-contained closed-system stability argument; the open/hole mismatch is a correctness gap, not a circular reduction.

full rationale

No circular step is present. Theorem A is proved by showing that any accumulation point mu0 of equilibrium states mu_n maximizes the closed pressure: the proof sets P_{f_n}(phi_n) := sup_{eta in Z_n} {h_eta(f_n)+int phi_n d eta}, obtains limsup P_{f_n}(phi_n) >= P_f(phi) via Lemma 3.0.3 (proved in the paper with a Brin-Katok entropy argument), and obtains h_{mu0}(f)+int(phi-P_f(phi)) d mu0 >= 0 via Lemma 3.0.1 (generating partition), giving equilibrium. These lemmas do not assume the conclusion. The cited inputs [34], [4], and [8] are prior finiteness/semicontinuity results whose assumptions do not include Theorem A, so they are real evidence rather than restatements. The clearest weakness is non-circular: the abstract and Section 2.4 call the systems 'open,' but F_Z contains no hole H and the proof never uses P_{f,H} or M_f(M,H); Theorem 2.3.1, cited from [34], is what supplies the open interpretation, and the paper does not establish P_{f,H}=sup_{Z(Lambda)} or stability of holes under C^0 perturbation. This is an omitted justification, not a reduction of the theorem to its inputs.

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

The paper introduces no numerical fitted parameters and no new postulated entities; its objects (zooming sets, special holes, inducing schemes, zooming potentials) are all defined or cited from prior work [31],[34],[3]. The central claim rests on cited prior theorems and on the proof lemmas listed in axioms.

assumptions (6)
  • domain assumption Theorem 2.2.1 ([34]): every zooming system with sufficiently small r0 and a special hole has a finite Markov structure adapted to the hole.
    Invoked in Section 2.2 to define the class of open systems; the proof is not reproduced.
  • domain assumption Theorem 2.3.1 ([34], Theorem B): for backward separated measurable open zooming systems with Lipschitz contractions and locally Hölder induced potentials, there are finitely many ergodic equilibrium states.
    Invoked in Section 2.3 and used as the finiteness input for Theorem C.
  • domain assumption Araujo [8][Theorem 11] applies to zooming systems and gives a generating partition P with limsup h_{µn}(f_n) ≤ h_{µ0}(f,P).
    Used in Lemma 3.0.1; the paper only sketches why the zooming contraction property satisfies the theorem's hypotheses.
  • ad hoc to paper C^0 convergence f_n → f preserves the family F_Z and the associated zooming sets and measures, and the holes, if present, can be handled consistently.
    The proof of Theorem A silently assumes this; it is not stated or proved, and the proof never uses the hole.
  • standard math Brin-Katok local entropy formula and the Ergodic Decomposition Theorem apply to the measures and maps considered in Lemma 3.0.3.
    Standard results used without proof in Section 3.
  • standard math Ledrappier-Walters relative variational principle (Theorem 4.0.1) and the fact that uniform contractions on fibers have zero fiber entropy.
    Used in Section 4 to reduce the skew-product potential to the base.

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Pith. "Pith review of Equilibrium Stability for Open Zooming Systems." pith.science (2026). https://pith.science/paper/FQKUHECI

@misc{pith2026250208693,
  author       = {Pith},
  title        = {Pith review of: Equilibrium Stability for Open Zooming Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FQKUHECI}},
  note         = {Machine review of arXiv:2502.08693}
}
read the original abstract

We prove that for a wide family of open zooming systems and zooming potentials we have equilibrium stability, i.e., the equilibrium states depend continuously on the dynamics and the potential. We consider the open zooming systems with special holes and quite general contractions and zooming potentials with locally H\"older induced potential, which include the H\"older ones. We also prove stability for skew-products with the base being a zooming system like above. As a consequence of finiteness and stability, we obtain uniqueness of equilibrium state.

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