Induced Topological Pressure for Dynamical Systems

Hanjing Zhu; Wenhui Ma; Yun Zhao

Induced Topological Pressure for Dynamical Systems

T0 review · 0 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that nonlinear induced topological pressure satisfies a variational principle relating it to entropy and potential integrals over invariant measures.

desk verdict Competent extension of induced pressure to the nonlinear setting; the new variational principle is real and the proof holds, but the main theorem is conditional on a strong hypothesis the paper never illustrates. read the letter →

arxiv 2507.07782 v1 pith:RECYDZFB submitted 2025-07-10 math.DS

Wenhui Ma , Yun Zhao , Hanjing Zhu This is my paper · ORCID

classification math.DS MSC 37D3528D2037A35

keywords inducedtopologicalpressurevariationalprinciplenonlinearthermodynamicformalismequilibriumstatefreezingsubdifferentialentropyergodicmeasures

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 introduces a high-dimensional nonlinear version of induced topological pressure, $P^F_\psi(\Phi)$, which counts orbit complexity at time scales selected by a positive continuous function $\psi$ while weighting a vector of observables $\Phi$ through a continuous function $F$. The main result is a variational principle: if either $F$ is convex or the augmented system $(f,(\Phi,\psi))$ has an abundance of ergodic measures, then $P^F_\psi(\Phi)=\sup_{\nu\in M(X,f)} (h_\nu(f)+F(\int\Phi\,d\nu))/\int\psi\,d\nu$. For $d=1$ and $F=\mathrm{id}$ this recovers the known induced-pressure variational principle, and with $\psi\equiv1$ it reduces to the classical topological-pressure variational principle. The paper also proves properties of the classical induced pressure, including equilibrium states, subdifferentials, and a freezing-state criterion.

What carries the argument

The load-bearing construction is the $\psi$-induced partition of time: for each horizon $T$, the set $S_T=\{n:\exists x,\ S_n\psi(x)\le T<S_{n+1}\psi(x)\}$ records the relevant return times, and $X_n=\{x:S_n\psi(x)\le T<S_{n+1}\psi(x)\}$ are the corresponding orbit strips. The nonlinear induced pressure is built from these by summing $\exp(nF(S_n\Phi(x)/n))$ over $(n,\varepsilon)$-spanning or separated sets inside each $X_n$ and taking $\lim_{\varepsilon\to0}\limsup_{T\to\infty}\frac1T\log(\cdot)$. The proof's pivot is the augmented vector $\Psi=(\Phi,\psi)$ together with the one-parameter family $G_\beta(a,b)=F(a)-\beta b$, since the induced pressure turns out to be the crossing point where the ordinary nonlinear pressure $P^{G_\beta}(\Psi)$ changes sign.

What would settle it

Take a non-convex $F$, for example $F(u,v)=-(u-v)^2$, on a system whose ergodic measures are too sparse to approximate the invariant measures entering the supremum; compute $P^F_\psi(\Phi)$ from the defining spanning-set limit and compare it with the right-hand side of Theorem 4.6. Any strict gap between the two numbers would refute the variational principle.

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Extended reading notes

Core claim

The paper's central claim is Theorem 4.6. For a continuous map on a compact metric space, a positive continuous scaling function $\psi$, a vector of potentials $\Phi=(\varphi_1,\dots,\varphi_d)$, and a continuous $F:\mathbb{R}^d\to\mathbb{R}$, the high-dimensional nonlinear induced topological pressure satisfies $$P^F_\psi(\Phi)=\sup_{\nu\in M(X,f)}\frac{h_\nu(f)+F(\int\Phi\,d\nu)}{\int\psi\,d\nu}$$ whenever either $F$ is convex or $(f,\Psi)$ with $\Psi=(\Phi,\psi)$ has an abundance of ergodic measures. The bridge is Corollary 4.2: with $G_\beta(a,b)=F(a)-\beta b$, the induced pressure is the common infimum and supremum of the set of $\beta$ where the ordinary nonlinear pressure $P^{G_\beta}(\Psi)$ is non-positive or non-negative, so $P^F_\psi(\Phi)$ is the root of the map $\beta\mapsto P^{G_\beta}(\Psi)$. Feeding the known nonlinear variational principle for $P^{G_\beta}$ into that root equation produces the quotient formula.

Load-bearing premise

The non-convex half of Theorem 4.6 rests on the 'abundance of ergodic measures' hypothesis for the augmented vector, and the paper gives no concrete system known to satisfy it; if that hypothesis fails and $F$ is not convex, the quotient formula has no proof and can fail.

Editorial extensions

If this is right

For convex $F$, the variational principle holds for every topological dynamical system, so the induced pressure is obtained directly from invariant measures without extra dynamical hypotheses.
Through Corollary 4.2, the induced pressure can be located by finding the zero of $\beta\mapsto P^{G_\beta}(\Psi)$, reducing one new thermodynamic quantity to an existing one.
For $d=1$ and $F=\mathrm{id}$, all the Section 2 and Section 3 results apply to the classical induced pressure $P_\psi(\varphi)$, including open-cover definitions, convexity, continuity, and cocycle invariance.
Equilibrium states of the induced pressure form a convex set whose extreme points are ergodic, and a potential freezes exactly when the pressure curve becomes affine with slope $\mathrm{Max}_\psi(\varphi)$ and intercept $h^\psi_\infty(\varphi)$ (Theorem 3.4).
The nonlinear induced pressure is invariant under topological conjugacy after pulling back $\Phi$ and $\psi$, matching the invariance pattern of the ordinary nonlinear pressure (Theorem 4.4).

Reading between the lines

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

If the abundance condition is genuinely hard to verify, the convex branch is the operative version of Theorem 4.6, and concrete systems satisfying the non-convex branch would substantially widen its reach.
The root characterization suggests a practical numerical route: approximate $P^{G_\beta}(\Psi)$ for a range of $\beta$ and bisect, which would compute induced pressure even when the variational formula is not covered by the theorem.
The freezing-state criterion of Section 3 should carry over to the nonlinear family $P_\psi(\beta\Phi)$, since Theorem 4.6 makes that one-parameter family available and zero-temperature limits would then select maximizing measures of the normalized nonlinear potential.
Because the construction only requires a positive scaling function and pressure machinery, the same induced-pressure scheme may extend to flows or random dynamical systems, a direction the paper does not pursue.

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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

Dear colleague,

The paper does one genuinely new thing: it defines a high-dimensional nonlinear induced topological pressure (Section 4.2) and proves a variational principle for it (Theorem 4.6). That result is not in the cited work of Xing–Chen, Barreira–Holanda, or Hedges. The proof strategy is sensible: relate the new pressure to the root of an ordinary nonlinear pressure (Corollary 4.2), then invoke the variational principle from Barreira–Holanda. I checked the main line and it holds. The earlier sections, on equivalent definitions, equilibrium states, and freezing states for the classical induced pressure, are mostly routine extensions of existing arguments. They are competently done.

The weakest point is the abundance hypothesis in Definition 4.1. The paper gives no example of a system satisfying it, and it is a nontrivial condition on the pair (f, Ψ). The convexity branch of Theorem 4.6 avoids it, so the result is not hostage to an unverifiable assumption. But the strongest form of the theorem is conditional on a hypothesis the authors never unpack. That is a real presentational gap, not a mathematical error.

Minor issues: an undefined 'm' in the proof of Theorem 4.5, and a reversed inequality sign in the middle of the proof of Theorem 4.6 (the '0 >' should be '0 ≥' or '≤'; the conclusion is unaffected). The reader's report flagged a flawed spanning-set argument in Theorem 2.3(vi). I do not see a real flaw there — for separated sets the inequality Σ a_x b_x ≤ (Σ a_x)(Σ b_x) gives P(φ+g) ≤ P(φ)+P(g) directly. The text uses spanning sets and writes the product bound for sums over n, which is slightly sloppy, but the statement is true and the proof is repairable.

The citation practice is honest. The new part builds explicitly on [3] and [28], and self-citation is not an issue. There is no fitting of data or circular reasoning.

Bottom line: this is a solid, incremental paper. It will be useful to people working in thermodynamic formalism, multifractal analysis, and large deviations. It deserves a serious referee. I would send it out, with the expectation of minor revision on exposition and hopefully an example for the abundance condition.

Referee Report

0 major / 6 minor

Summary. The paper studies induced topological pressure in two settings. For the classical induced pressure introduced by Xing and Chen, it provides equivalent open-cover definitions and elementary properties such as monotonicity, continuity, convexity, bounds, and a characterization of invariant measures. It then develops equilibrium states, subdifferentials, and freezing/zero-temperature states for this pressure, extending earlier work of Hedges. The second half introduces a high-dimensional nonlinear induced topological pressure, proves a root representation in terms of the nonlinear pressure of an augmented potential (Theorem 4.5), and establishes a variational principle (Theorem 4.6) under either an abundance-of-ergodic-measures condition or convexity of the nonlinearity F. The main result unifies the classical induced-pressure variational principle and the higher-dimensional nonlinear pressure of Barreira and Holanda.

Significance. If the results are correct, the paper gives a useful unified formalism: the nonlinear induced pressure is shown to equal the supremum of (hν + F(∫Φ dν))/∫ψ dν over invariant measures, with the root characterization of Theorem 4.5 as an intermediate tool. The proofs are honest and transparent: the central derivation relies on the cited variational principles of Xing-Chen and Barreira-Holanda, and I found no circularity or parameter-fitting. The abundance condition in Definition 4.1 is indeed strong, but it is an explicit hypothesis rather than a hidden assumption; the paper would be easier to use with examples, and I do not regard the absence of examples as a correctness defect. The main chain from Theorem 4.5 through Corollary 4.2 to Theorem 4.6 is logically coherent, and the secondary material on equilibrium states and freezing states is mostly sound, with one local proof gap noted below.

minor comments (6)

[Theorem 4.5] The symbol m is used throughout the proof, for instance in the intervals (ℓ−1)m < Snψ(x) ≤ ℓm, but it is never defined in the statement. From the context it must be m = inf ψ; it should be introduced explicitly before the proof.
[Theorem 4.6 proof] In the upper-bound half, the displayed line '0 > (hν(f)+F(∫Φdν))/∫ψdν − β' should read '0 ≥ ...'. The intended conclusion is unaffected, but as written the inequality does not follow from the preceding line P^{Gβ}(Ψ) ≤ 0.
[Corollary 3.2] The proof invokes Theorem 3.3, but Theorem 3.3 requires a subset F to be the full set of equilibrium states and requires property (2) to hold for all members of F. The present proof does not show that {μ} is the entire equilibrium set, nor that every equilibrium state maximizes ∫ϕ/∫ψ. The corollary is nevertheless true by a direct argument using hν+∫ϕ/∫ψ ≤ Pψ(ϕ) for all ν, so the proof should be rewritten accordingly.
[Corollary 4.2] The continuity of the map β ↦ P^{Gβ}(Ψ), which is needed for the equality inf{β : P^{Gβ}(Ψ) ≤ 0} = sup{β : P^{Gβ}(Ψ) ≥ 0}, is asserted without proof. It follows from the continuity of nonlinear pressure in the potential, but the relevant estimate should be stated explicitly.
[Definition 4.1 and Theorem 4.6] The abundance condition is strong and no concrete system satisfying it is mentioned. A remark giving standard examples, such as mixing subshifts of finite type where ergodic measures are entropy dense, would significantly improve the paper's usability and make clear that the hypothesis is non-vacuous.
[Throughout] There are several small typographical and notational issues: in Theorem 4.4 the set SY_T is defined with 'y ∈ X' instead of 'y ∈ Y'; Corollary 4.1 writes lim_{T→∞} where limsup_{T→∞} is meant; Corollary 3.1 contains 'Morevoer'; the proof of Theorem 4.3 uses the notation 'P ψ F' in place of 'P^F_ψ'; and Remark 4.2 delegates an open-cover equivalence to 'similarly' without a sketch. These should be corrected in a final pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the variational principle is built from independent external theorems (Xing–Chen and Barreira–Holanda) plus the paper's own technical estimates.

full rationale

The paper's central claims are not circular. The classical induced-pressure variational principle (Theorem 2.2) is quoted from [28], and the nonlinear variational principle (Theorem 4.1) is quoted from [3]; neither is the authors' own prior result, and neither depends on the conclusions of this paper. The new object, the nonlinear induced pressure P^F_psi(Phi), is defined directly from spanning/separated sums in Section 4.2, and its relation to the nonlinear pressure P^{G_beta}(Psi) is proved in Theorem 4.5 and Corollaries 4.1–4.2 using the paper's own estimates. Theorem 4.6 then combines Corollary 4.2 with Theorem 4.1 by a standard root-location argument to obtain the variational principle. There is no fitted parameter later renamed as a prediction, no definition of the target quantity in terms of itself, and no load-bearing self-citation: the authors do not rely on any prior work by themselves, and the cited results are external mathematical theorems that do not presuppose the statement being proved. The 'abundance of ergodic measures' hypothesis in Definition 4.1 is imported from [3] as an explicit assumption; being strong or hard to verify is a non-vacuousness concern, not a circularity. The proof of Theorem 4.6 contains a sign typo ('0 >' should be '0 ≥' in the first half), and Theorem 4.5 uses an undefined m in the paragraph after (4.5), but these are presentational issues that do not make the derivation circular. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities are introduced; the work is purely mathematical, and all constants are fixed by the data (φ, ψ, F, f). The main new object, nonlinear induced pressure, is a definition, not an entity needing independent evidence.

assumptions (4)

standard math Variational principle for classical induced topological pressure ([28, Corollary 3.2])
Used without proof in Theorem 2.3(ii) and throughout Section 2; it is an external result from Xing-Chen.
standard math Variational principle for high-dimensional nonlinear topological pressure ([3, Theorem 1.1])
Quoted as Theorem 4.1 and used as a black box in Corollary 4.2 and Theorem 4.6.
domain assumption Abundance of ergodic measures for (f, Ψ) is assumed in one branch of Theorem 4.6
This condition on the dynamics is taken from [3]; the paper neither verifies nor constructs systems satisfying it, so the theorem is conditional on it.
domain assumption ψ continuous with 0 < m ≤ ψ ≤ M and F continuous
Explicitly stated in the definitions (Section 2, Section 4.2); positivity of ψ is essential for the time-change and the finiteness of S_T.

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Cite this review

Pith. "Pith review of Induced Topological Pressure for Dynamical Systems." pith.science (2026). https://pith.science/paper/RECYDZFB

@misc{pith2026250707782,
  author       = {Pith},
  title        = {Pith review of: Induced Topological Pressure for Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RECYDZFB}},
  note         = {Machine review of arXiv:2507.07782}
}

read the original abstract

This paper is devoted to the study of induced topological pressure, including both classical and nonlinear cases. For the classical induced topological pressure, we investigate equilibrium states, subdifferential and freezing states, while also discussing some basic properties of it. Additionally, the high dimensional nonlinear induced topological pressure is introduced, and the corresponding variational principle is established.

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [3]

Barreira, C

L. Barreira, C. Holanda, Higher-dimensional nonlinear thermodynamic formalism, J. Stat. Phys., 2022, 187(2): 18, 28pp

work page 2022

[2] [28]

Z. Xing, E. Chen, Induced topological pressure for topological dynamical systems, J. Math. Phys., 2015, 56(2): 022707, 10 pp

work page 2015

[3] [1]

R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 1965, 114: 309-319

work page 1965

[4] [2]

Barreira, Thermodynamic formalism and applications to dimension theory, Progr

L. Barreira, Thermodynamic formalism and applications to dimension theory, Progr. Math., 294, Birkh¨ auser/Springer Basel AG, Basel, 2011

work page 2011

[5] [4]

Barreira, C

L. Barreira, C. Holanda, Nonlinear thermodynamic formalism for flows, Dynamical Systems, 2022, 37(4): 603-629

work page 2022

[6] [5]

Barreira, J

L. Barreira, J. Schmeling, Sets of non-typical points have full topological entropy and full Hausdorff dimension, Israel Journal of Mathematics, 2000, 116(1): 29-70

work page 2000

[7] [6]

Bowen, Topological entropy for noncompact sets, Trans

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 1973, 184: 125-136

work page 1973

[8] [7]

Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470, Springer Berlin Heidelberg, 1975

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470, Springer Berlin Heidelberg, 1975

work page 1975