Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem

Thiago Bomfim; Victor Carneiro

REVIEW 3 major objections 5 minor 37 references

Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a broad class of positively expansive maps, a dense set of smooth potentials has no phase transition at any inverse temperature.

desk verdict Theorems A and B are a genuine higher-dimensional extension and look plausible, but Theorem C's proof relies on a weak-contracting hypothesis that fails for indifferent breakpoints, so the headline application is unsupported. read the letter →

arxiv 2505.23934 v1 pith:KNJ74MAB submitted 2025-05-29 math.DS

Thiago Bomfim , Victor Carneiro This is my paper

classification math.DS MSC 82B2637D3537C3037C40

keywords phasetransitionspectralgappropertytransferoperatorpositivelyexpansivedynamicsintermittentskew-productsexpandingonaveragepotentialsthermodynamicformalismHölder

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

Uniformly expanding maps have no phase transitions for Hölder potentials because their transfer operators have a spectral gap. This paper pushes the same conclusion toward dynamics that are only positively expansive, including skew-products whose circle fibers are intermittent maps with indifferent points. The authors establish that a dense set of smooth potentials is expanding on average, which forces the transfer operator to be quasi-compact and hence spectrally gapped, and they upgrade this to all parameters $t\in\mathbb{R}$ in the intermittent skew-product setting. The central result, Theorem C, says that for a class of intermittent skew-products over expanding or intermittent torus maps, a dense family of smooth potentials makes $\mathcal{L}_{f,t\varphi}$ spectrally gapped on every Hölder space $C^\alpha$ for every real $t$, so no phase transition occurs for those potentials. The result matters because non-uniformly hyperbolic systems are exactly where geometric potentials are known to produce phase transitions; showing that smooth potentials generically avoid them locates the mechanism in the potential rather than in the dynamics.

What carries the argument

The engine is the transfer operator $\mathcal{L}_{f,\varphi}(g)(x)=\sum_{f(y)=x}e^{\varphi(y)}g(y)$: when it has a spectral gap on a Banach space, the topological pressure equals the logarithm of the leading eigenvalue and the pressure function is analytic in $t$. To find potentials with a spectral gap, the paper introduces expanding-on-average potentials, those whose pressure strictly exceeds the supremum over invariant measures that are not expanding on average; a sharp essential-spectral-radius estimate turns this property into quasi-compactness and then a spectral gap. Density of such potentials comes from an ergodic-optimization theorem for invariant measures with entropy above a fixed threshold. For Theorem C, the extra mechanism is a flat-potential criterion for weakly contracting uniform backward random walks: if the walk contracts distances and the potential's orbit sums are controlled along the natural coupling of preimage branches, then the potential is flat and the transfer operator is spectrally gapped. The proof constructs a dense class of flat potentials that are constant in the circle direction near each indifferent fiber breakpoint and in the torus direction near each base breakpoint, so the slow contraction near indifferent points is neutralized.

What would settle it

Take an intermittent circle fiber map with derivative exactly 1 at a fixed point and constant breakpoints, let $F(x,y)=(g(x),f(y))$ with $g$ expanding, and feed the natural coupling of $F^{-m}$ two points approaching that fixed point in the fiber; the preimage branch through the fixed point keeps their distance essentially unchanged, contradicting the uniform-contraction premise behind Theorem C. A direct numerical computation of the spectral radius of $\mathcal{L}_{F,t\varphi}$ on $C^\alpha$ for a smooth flat potential would then show whether the conclusion survives.

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Extended reading notes

Core claim

The paper's central claim is that typical smooth potentials produce the strongest form of thermodynamic regularity in a setting that includes non-uniformly hyperbolic dynamics. Theorem A says that for every positively expansive $C^r$ local diffeomorphism of a compact connected manifold with at least one repeller periodic point, an open dense set of continuous potentials has the spectral gap property for the transfer operator on $C^r$, and also for all sufficiently large and all sufficiently small $|t|$; the complement is nowhere dense. Theorem B says that for skew-products $F(x,y)=(g(x),f_x(y))$ with expanding or intermittent torus base and constant fiber breakpoints (classes $TM_2$ and $TM_3$), every smooth potential has analytic pressure on an open dense set of parameters $t$. Theorem C, the strongest statement, says that in the broader class $TM_1$ (base $g$ expanding or intermittent, with fiber breakpoints allowed to depend on $x$) there is a dense set $\widetilde{H}$ of smooth potentials for which $\mathcal{L}_{F,t\varphi}$ has the spectral gap on every Hölder space $C^\alpha$ for every real $t$, and consequently $F$ has no phase transition with respect to $\varphi$.

Load-bearing premise

The load-bearing premise is that, for a high iterate of the skew-product, any two nearby points have preimage sets that can be paired so that every pair of preimages is closer than the original points and at least one pair is closer by a fixed percentage; near fixed points where the derivative is exactly 1 this can fail, since the relevant preimages are only infinitesimally closer.

Editorial extensions

If this is right

For any positively expansive $C^r$ local diffeomorphism with a repeller periodic point, a dense, open set of smooth potentials has no low-temperature or high-temperature phase transition.
For intermittent skew-products in $TM_2$ or $TM_3$, every smooth potential has analytic pressure on an open dense parameter set, so any phase transition is confined to a small exceptional set of inverse temperatures.
In the $TM_1$ class, a dense family of flat smooth potentials makes $\mathcal{L}_{F,t\varphi}$ spectrally gapped on every Hölder space for every real $t$; those potentials have no phase transition at all.
The set of smooth potentials for which the spectral gap fails is nowhere dense in the uniform topology, so failure is topologically rare.
Whenever the spectral gap holds, the corresponding equilibrium state is unique and the pressure is strictly convex when the potential is not cohomologous to a constant.

Reading between the lines

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

If Theorem C is right, all standard consequences of a spectral gap—exponential decay of correlations, large-deviation principles, and analyticity of thermodynamic quantities—hold for the dense flat potentials at every real $t$; the paper does not spell these out, but they follow from the operator-theoretic setup.
The proof's uniform-contraction premise suggests a testable sharpening: replacing uniform contraction by a controlled slow-contraction bound along the indifferent set might extend the all-$t$ conclusion to a wider class of positively expansive maps with indifferent periodic points.
The known phase transitions for geometric potentials such as $-\log|f'|$ in intermittent maps involve potentials that are not flat in the paper's sense, so the density result and the known counterexamples can coexist; a natural next question is whether every smooth potential is a limit of potentials with no phase transition.
Theorem B's open-dense parameter set leaves open whether the exceptional parameters can be removed; answering that would settle the paper's Question C about density of potentials with spectral gap for all real $t$.

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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

Two things to know. The genuinely new content is Theorem A, extending the density of the spectral gap property from transitive circle local diffeomorphisms to positively expansive C^r local diffeomorphisms on compact connected manifolds with a repeller periodic point, and Theorem B, giving analyticity of pressure on a dense open set of parameters for intermittent skew-products in classes TM2/TM3. The proofs of A and B are assembled from earlier machinery — BC23's transfer operator lemmas, CL97's essential spectral radius estimates, LT24's ergodic optimization — and the adaptation to higher dimensions looks legitimate. The repeller periodic point assumption is used honestly, and the open questions at the end are reasonable.

The soft spot is Theorem C. The proof invokes [KL20, Theorem 5.8] via the weak contracting condition in Definition 4.11. That requires a contraction function c(r) < r for all r > 0, plus a uniformly contracting pair with ratio at most 1/λ. The paper asserts that for large m, the inverse branches of F^m can be matched so one pair contracts by a uniform σ < 1 and the others are strictly closer. That uniform σ fails when the intermittent factor has an indifferent breakpoint: an inverse branch through the indifferent point has derivative approaching 1, so the distance ratio between matched preimages tends to 1 as the points approach the breakpoint. Pointwise strict inequality for finite j does not produce a global contraction function. The remark about expansion away from breakpoints only handles pairs that stay away from the indifferent sets; it does not control matchings that pass through them. So the hypotheses of [KL20, Theorem 5.8] are not established, and Theorem C does not follow from the argument given.

I don't want to overstate the damage. Theorems A and B may well be correct, and there is no circularity or data fitting; the self-citations to BC23 and BF23 are legitimate support. The flaw is concentrated in Section 4.6, but it is load-bearing for a headline claim. A serious revision would need to either prove a suitable contraction property away from indifferent breakpoints or replace Theorem C with a weaker statement. The paper is for specialists in thermodynamic formalism; the setting is new and the machinery is standard. I would send it to peer review, with a clear instruction that the referee focus on Section 4.6 and the validity of the weak-contracting hypothesis for intermittent maps.

Referee Report

3 major / 5 minor

Summary. The paper studies positively expansive C^r local diffeomorphisms on compact manifolds, aiming to extend one-dimensional results on the density of potentials for which the transfer operator has the spectral gap property and for which phase transitions are absent. Theorem A establishes, for any such map with a repeller periodic point, an open and dense set of continuous potentials such that all sufficiently large and sufficiently small multiples of the potential yield a transfer operator with the spectral gap property on C^r. Theorem B claims that for intermittent skew-products of types TM2 and TM3, every smooth potential has an open dense set of parameters t for which the pressure function t ↦ P_top(F,tφ) is analytic. Theorem C claims that for the broader class TM1, there is a dense set of smooth potentials for which F has no phase transition and the transfer operator L_{F,tφ} has the spectral gap property on C^α for all t ∈ R and all 0 < α < 1. The proofs combine ergodic optimization, expanding-on-average potentials, and Kloeckner's optimal-transport criterion for spectral gap.

Significance. If the results were correct, they would constitute a meaningful higher-dimensional extension of the circle-map theory of Bomfim–Carneiro and Bomfim–Fernandes, and would provide the first generic absence-of-phase-transition statement for a class of intermittent skew-products. The paper has clear strengths: it identifies a natural class of positively expansive local diffeomorphisms, exploits the conjugacy to expanding maps, and uses the modern machinery of ergodic optimization and flatness of potentials. Theorems A and B appear plausible and, modulo the unproved transfer of lemmas from [BC23], may be correct. However, the central new result, Theorem C, rests on an assertion about uniform contraction of inverse branches that fails for maps with indifferent breakpoints. Since the paper's main advertised application is precisely to intermittent skew-products, this gap undermines the principal contribution.

major comments (3)

[Section 4.6, Proof of Theorem C] The proof begins with the assertion that for F̃ = F^m there exists 0 < σ < 1 such that the preimages of any two points z1,z2 can be ordered to satisfy d(w_{1,1},w_{2,1}) ≤ σ d(z1,z2) and d(w_{1,j},w_{2,j}) < d(z1,z2) for all j ≥ 2. For an intermittent factor with an indifferent breakpoint α (|f'_x(α)| = 1), the inverse branch through α has derivative arbitrarily close to 1 as the point approaches α, so the contraction ratio tends to 1. Hence no uniform σ < 1 can exist, and the uniform backward random walk of F^m is not weakly contracting in the sense of Definition 4.11: there is no contraction function c(r) < r and no λ > 1 satisfying item (ii) of that definition. Consequently Lemma 4.14 and Theorem 4.15 from [KL20] cannot be applied, and the spectral gap conclusion for L_{F,tφ} on C^α does not follow. This is a load-bearing error for Theorem C.
[Section 4.6, Proof of Theorem B] The proof that the set A is dense is incomplete. The text asserts: 'let t̃ ∉ A and ε>0. Then there exists |t̃̃−t̃|<ε and j such that P_top(g,t̃̃φ(·,α_j)) > max{...} or P_top(f_{x_j},t̃̃φ(x_j,·)) > max{...}.' No justification is given for the existence of such a nearby parameter with strict dominance. Without this, the density of A is not established. Moreover, the definition A = A1 ∪ ∪_j int(A2,j) ∪ ∪_j int(A3,j) does not exclude the at-most-two exceptional parameters in each int(A2,j) or int(A3,j) where, according to [BF23], the corresponding circle pressure may fail to be analytic; therefore the claimed analyticity of t ↦ P_top(F,tφ) on all of A is not justified as written.
[Section 3.3] Lemma 3.1, Corollary 3.2, Lemma 3.4, and Proposition 3.5 are imported from [BC23] with the statement that 'the same proofs as in [BC23]' work in the higher-dimensional setting. These results are load-bearing for Theorems A and B, and it is not self-evident that the one-dimensional arguments carry over unchanged; for example, the proof of Lemma 3.1 may rely on circle-specific properties such as the structure of inverse branches or one-dimensional coboundary equations. The authors should either provide the proofs adapted to the present setting or explain in detail why the dimension is irrelevant.

minor comments (5)

[Section 2.1, Example 2.5] The class TM1 includes the case 'g is intermittent', but intermittent maps are defined only on S^1 (Definition 2.3), while g is a map on T^d. The paper later (Section 4.6) implicitly assumes d=1 when g is intermittent. The statement of Theorem C should clarify the domain of g or restrict to d=1 in that case.
[Theorem C statement] Items (i) and (ii) use the lowercase letter f ('f has no phase transition', 'L_{f,tφ}') although the dynamics under consideration is F. This is a typo that should be corrected.
[Theorem A statement] The phrase 'for all |t| ≥ t0 or |t| ≤ t1' is ambiguous and should read 'for all |t| ≥ t0 and for all |t| ≤ t1', as is proved in Propositions 4.8 and 4.10.
[Definition 4.12] There is a duplicated phrase in the definition of the natural coupling: 'x1 = x_{η(j1)} ∈ T^{-1}(x) and x1 = x_{η(j1)} ∈ T^{-1}(x)' should be 'x1 = x_{η(j1)} ∈ T^{-1}(x) and y1 = y_{σ(j1)} ∈ T^{-1}(y)'.
[Proof of Theorem C, last paragraph] The sentence 'Since L_{F,φ} is a positive operator then there exists a probability ν such that L*_{F,φ}ν = ρ(L_{F,φ}|C0)ν' is stated without proof; while this is a standard consequence of the Schauder–Tychonoff theorem for positive operators on C(M), the justification should be given or referenced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral-gap claims are derived from external operator criteria and new density arguments, not from definitionally loaded inputs or fitted parameters.

full rationale

The derivation chain is not circular. Theorem A is obtained by defining expanding/expanding-on-average potentials through pressure and Lyapunov data, then invoking the essential spectral radius estimates of [CL97] and the ergodic-optimization genericity result [LT24]; the spectral gap, analyticity, and strict convexity conclusions are outputs, not inputs, of these definitions. Theorem B reduces any non-spectral equilibrium of the skew-product to a product measure concentrated on a breakpoint, yielding P(F,tφ) = P(g,tφ(·,α_j)) or P(f_{x_j},tφ(x_j,·)); the subsequent use of [BF23] for the one-dimensional fiber is a normal invocation of a prior theorem, while the genuinely new step is the skew-product reduction itself. Theorem C applies the externally stated weak-contraction and flatness criteria of [KL20] (Lemma 4.14 and Theorem 4.15) to conclude the spectral gap property for L_{F,tφ}; no parameter is fitted to a quantity later announced as a prediction, and no equation is defined in terms of the conclusion it is meant to establish. The self-citations to [BC23] and [BF23] supply background lemmas and motivational one-dimensional results whose assumptions do not include the present claims. The reviewer-level objection that the weak-contraction hypothesis in Definition 4.11 may fail at indifferent breakpoints of F^m is a mathematical correctness risk in applying [KL20, Theorem 5.8], not a circularity: it concerns whether an external theorem's hypotheses are satisfied, not whether the paper's conclusion is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on several background assumptions imported from prior literature, on an explicit hypothesis about repeller periodic points, and on one ad hoc assumption in the proof of Theorem C that appears to be false for intermittent dynamics. No free parameters are fitted to data, and no new entities are postulated.

assumptions (5)

domain assumption Every positively expansive C^r local diffeomorphism on a compact connected manifold is topologically conjugate to a uniformly expanding map.
Invoked in Section 3.1 and used to obtain specification, entropy density, upper semicontinuity, and generating partitions; cited from [CR80].
domain assumption f admits at least one repeller periodic point.
Explicit hypothesis in Theorem A; used in Lemma 4.6 and Proposition 4.10 to build expanding measures. The authors note in Question A that this may not follow from positive expansivity.
standard math The Campbell-Latushkin essential spectral radius estimates (Theorem 3.8) apply to C^r potentials and the transfer operator on C^r.
Quoted from [CL97] and used in Proposition 4.3; assumed valid for all smooth coverings on compact manifolds.
ad hoc to paper The lemmas from [BC23] (Lemma 3.1, Corollary 3.2, Lemma 3.4, Proposition 3.5) extend to higher-dimensional positively expansive maps with the same proofs.
Section 3.3 states this without proof; these lemmas are used to identify pressure with log spectral radius and obtain analyticity.
ad hoc to paper The uniform backward random walk of F^m is weakly contracting (there exists σ < 1 and a contraction function c(r) < r).
Asserted in the proof of Theorem C (Section 4.6) to apply [KL20]; false for maps with indifferent fixed points because inverse branches through indifferent points have contraction ratio tending to 1.

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Pith. "Pith review of Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem." pith.science (2026). https://pith.science/paper/KNJ74MAB

@misc{pith2026250523934,
  author       = {Pith},
  title        = {Pith review of: Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KNJ74MAB}},
  note         = {Machine review of arXiv:2505.23934}
}

abstract

It is known that all uniformly expanding dynamics $f: M \rightarrow M$ have no phase transition with respect to a H\"older continuous potential $\phi : M \rightarrow \mathbb{R}$, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f , t\phi)$ is analytical. Moreover, the associated transfer operator $\mathcal{L}_{f , t\phi}$, acting on the space of H\"older continuous functions, has the spectral gap property for $t \in \mathbb{R}$. For dynamics that are topologically conjugate to an expanding map, a full understanding has yet to be achieved. On the one hand, by \cite{KQW21,KQ22}, for such maps and continuous potentials, the associated topological pressure function can behave wildly. On the other hand, by \cite{BF23}, for transitive local diffeomorphisms on the circle and a large class of H\"older continuous potentials, the phase transition does not occur, and the associated transfer operator has the spectral gap property for all parameters $t \in \mathbb{R}$. As a first approach to understanding what happens in high dimensions, in this paper, we study positively expansive local diffeomorphisms. In particular, we show that the associated transfer operator has the spectral gap property for a large class of regular potentials. Moreover, for a class of intermittent skew-products and a large class of regular potentials, we obtain phase transition results analogous to \cite{BF23}.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

V. Baladi. Positive transfer operators and decay of correlations. World Scientific Publishing Co. Inc. , 2000

work page 2000

[2] [2]

Bomfim and V

T. Bomfim and V. Carneiro. Thermodynamical and spectral phase transition for local diffeomorphisms in the circle. Nonlinearity , 36 (12), 2023

work page 2023

[3] [3]

Bomfim, A

T. Bomfim, A. Castro and P. Varandas. Differentiability of thermodynamical quantities in non-uniformly expanding dynamics. Advances in Math. 292, 478-528, 2016

work page 2016

[4] [4]

Bomfim and A

T. Bomfim and A. Castro. Linear Response, and Consequences for Differentiability of Statistical Quantities and Multifractal Analysis. J Stat Phys , 174, 135-159, 2019

work page 2019

[5] [5]

Bomfim and A

T. Bomfim and A. Fernandes. Phase transitions for transitive local diffeomorphism with break points on the circle and Holder continuous potentials. Preprint Arxiv , 2023

work page 2023

[6] [6]

R. Bowen. Equilibrium states and the ergodic theory of A nosov diffeomorphisms. Lect. Notes in Math. , volume 470, Springer Verlag, 1975

work page 1975

[7] [7]

Bowen and D

R. Bowen and D. Ruelle. The ergodic theory of Axiom A flows. Invent. Math. , 29, 181-202, 1975

work page 1975

[8] [8]

Campbell and Y

J. Campbell and Y. Latushkin. Sharp Estimates in Ruelle Theorems for Matrix Transfer Operators. Commun. Math. Phys. , 185, 379-396, 1997

work page 1997