Variational principle for random pressure function

Ercai Chen; Rui Yang; Xiaoyao Zhou

arxiv: 2210.13126 · v6 · pith:YEWBPA2Wnew · submitted 2022-10-24 · 🧮 math.DS

Variational principle for random pressure function

Rui Yang , Ercai Chen , Xiaoyao Zhou This is my paper

Pith reviewed 2026-05-24 10:39 UTC · model grok-4.3

classification 🧮 math.DS

keywords random pressure functionsvariational principlerandom dynamical systemsRuelle metric entropyKifer topological pressureconvex analysisergodic theory

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The pith

Random pressure functions for random dynamical systems satisfy a variational principle with Ruelle metric entropy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines random pressure functions for random dynamical systems by distilling the core properties of Kifer's topological pressure, and it defines Ruelle's metric entropy on invariant measures. Applying methods from convex analysis together with ergodic theory, the authors prove that each random pressure function equals the supremum, over all invariant measures, of the sum of the measure's Ruelle entropy and the integral of the potential. The resulting equality supplies a direct link between the measure-theoretic and topological sides of random dynamics. From this single principle the paper derives variational statements for polynomial topological entropy, mean dimension, and preimage entropy quantities in the indicated classes of systems.

Core claim

By summarizing the fundamental properties of Kifer's topological pressure we introduce the concept of random pressure functions, and define Ruelle's metric entropy for invariant measures. Employing the techniques from convex analysis and ergodic theory, we establish a variational principle for random pressure functions. Consequently, this new variational principle allows us to establish a vital bridge between ergodic theory and topological dynamics. In particular, the variational principles for polynomial topological entropy in zero entropy systems, mean dimensions in infinite entropy systems, and preimage entropy-like quantities in non-invertible dynamical systems are obtained.

What carries the argument

Random pressure functions, obtained by extracting the essential properties of Kifer's topological pressure so that convex-analysis arguments apply directly.

If this is right

The variational principle supplies a bridge between ergodic theory and topological dynamics for random systems.
It yields a variational principle for polynomial topological entropy inside zero-entropy systems.
It yields a variational principle for mean dimension inside infinite-entropy systems.
It yields variational principles for preimage entropy-like quantities inside non-invertible systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same convex-analytic route might be tried on other pressure-type functionals that have not yet been treated variationally.
Explicit computation of the pressure via the variational formula could be tested on simple random maps such as random rotations or random interval maps.
The definition via fundamental properties may allow the principle to be ported to settings beyond the original Kifer framework.

Load-bearing premise

The random pressure functions admit the application of convex analysis techniques to produce the variational equality with Ruelle's metric entropy.

What would settle it

A concrete random dynamical system together with a random pressure function for which the pressure value differs from the supremum of Ruelle entropy plus integrated potential over all invariant measures.

read the original abstract

For random dynamical systems, by summarizing the fundamental properties of Kifer's topological pressure we introduce the concept of random pressure functions, and define Ruelle's metric entropy for invariant measures. Employing the techniques from convex analysis and ergodic theory, we establish a variational principle for random pressure functions. Consequently, this new variational principle allows us to establish a vital bridge between ergodic theory and topological dynamics. In particular, the variational principles for polynomial topological entropy in zero entropy systems, mean dimensions in infinite entropy systems, and preimage entropy-like quantities in non-invertible dynamical systems are obtained.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines random pressure functions from Kifer's pressure and proves a variational principle equating them to Ruelle metric entropy, which then recovers several other entropy variational principles as special cases.

read the letter

The core contribution is the introduction of random pressure functions that capture the key properties of Kifer's topological pressure for random dynamical systems, followed by a variational principle that identifies this pressure with Ruelle's metric entropy for invariant measures. The authors then use this to derive variational principles for polynomial topological entropy in zero-entropy systems, mean dimension in infinite-entropy systems, and preimage entropy quantities in non-invertible systems. This does unify several existing notions under one random-dynamical umbrella using standard convex-analysis and ergodic-theory tools, which is a clean organizational move if the definitions hold up. The abstract presents the work as an extension rather than a reduction to prior results, and the citation pattern looks standard for the subfield. The main limitation visible from the abstract is the lack of explicit definitions, assumptions on the random base, or proof sketches, so it is impossible to check whether the convex-analysis step applies without extra conditions or whether the random pressure functions are the most natural choice. If the full paper supplies those details cleanly, the unification is useful within random dynamics; if not, the bridge claim stays formal. This is specialized work aimed at researchers already working on entropy in random or non-invertible systems. A reader looking for new variational principles in that niche would find it worth reading. It deserves peer review so referees can verify the technical steps and the scope of the recovered results.

Referee Report

0 major / 0 minor

Summary. The paper introduces the concept of random pressure functions for random dynamical systems by summarizing the fundamental properties of Kifer's topological pressure, defines Ruelle's metric entropy for invariant measures, and uses techniques from convex analysis and ergodic theory to establish a variational principle relating the two. This principle is then applied to derive variational principles for polynomial topological entropy in zero-entropy systems, mean dimensions in infinite-entropy systems, and preimage entropy-like quantities in non-invertible systems.

Significance. If the central variational principle is rigorously established with the stated assumptions, the work would provide a useful bridge between ergodic theory and topological dynamics in the random setting, extending classical variational principles and yielding new results for several entropy notions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for acknowledging the potential utility of the variational principle as a bridge between ergodic theory and topological dynamics in the random setting. The recommendation is listed as uncertain, yet no specific major comments or technical concerns are raised in the report. We therefore interpret the uncertainty as a request for possible clarification rather than an indication of identified errors. Below we address the report point by point; because no concrete objections appear, the point-by-point section is empty.

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard external techniques to a defined object

full rationale

The paper defines random pressure functions by summarizing properties of Kifer's topological pressure, then applies techniques from convex analysis and ergodic theory to prove a variational principle equating it to Ruelle's metric entropy. No equations, lemmas, or steps in the provided abstract reduce the claimed result to a fitted parameter, self-definition, or self-citation chain. The central claim is presented as a consequence of independent mathematical tools applied to the new object, with no load-bearing self-references or renaming of known results indicated.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted from the text.

pith-pipeline@v0.9.0 · 5615 in / 1023 out tokens · 17298 ms · 2026-05-24T10:39:41.214393+00:00 · methodology

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Works this paper leans on

12 extracted references · 12 canonical work pages

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[YCZ22] R. Yang, E. Chen and X. Zhou, On variational princi- ple for upper metric mean dimension with potential, arXiv: 2207.01901. [YCZ23] R. Yang, E. Chen and X. Zhou, Some notes on variational principle for metric mean dimension, IEEE Trans. Inform. The- ory 69 (2023), 2796-2800. 1.School of Mathematical Sciences and Institute of Mathema tics, Nanjing ...

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[1] [1]

Bi´ s, M

[BCMP22] A. Bi´ s, M. Carvalho, M. Mendes and P. Varandas, A convex analysis approach to entropy functions, variational principles and equilibrium states, Comm. Math. Phys. 394 (2022), 215-

work page 2022

[2] [2]

Bogensch¨ utz, Entropy, pressure, and a variational princi- ple for random dynamical systems, Random Comput

[Bog92] T. Bogensch¨ utz, Entropy, pressure, and a variational princi- ple for random dynamical systems, Random Comput. Dynam. 1 (1992), 99-116. [CPV24] M. Carvalho, G. Pessil and P. Varandas, A convex analysis approach to the metric mean dimension: limits of scaled pres- sures and variational principles, Adv. Math. 436 (2024), Paper No. 109407, 54 pp. [C...

work page 1992

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Gromov, Topological invariants of dynamical systems an d spaces of holomorphic maps: I, Math

[Gro99] M. Gromov, Topological invariants of dynamical systems an d spaces of holomorphic maps: I, Math. Phys, Anal. Geom. 4 (1999), 323-415. [GLT16] Y. Gutman, E. Lindenstrauss and M. Tsukamoto, Mean di- mension of Zk actions, Geom. Funct. Anal. 26 (2016), 778-817. 26 [GS20] Y. Gutman and A. ´Spiewak, Metric mean dimension and ana- log compression, IEEE ...

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Gutman and A

[GS21] Y. Gutman and A. ´Spiewak, Around the variational principle for metric mean dimension, Studia Math. 261 (2021), 345-360. [GT20] Y. Gutman and M. Tsukamoto, Embedding minimal dynami- cal systems into Hilbert cubes, Invent. Math. 221 (2020), 113-

work page 2021

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Kifer, On the topological pressure for random bundle tra ns- formations, Amer

[Kif01] Y. Kifer, On the topological pressure for random bundle tra ns- formations, Amer. Math. Soc. Transl. Ser. 2 202 (2001), 197-

work page 2001

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Ledrappier and P

[L W77] F. Ledrappier and P. Walters, A relativised variational prin- ciple for continuous transformations, J. London Math. Soc. 16 (1977), 568-576. [Li13] H. Li, Soﬁc mean dimension, Adv. Math. 244 (2013), 570-604. [Lia22] B. Liang, Conditional mean dimension, Erg. Theory Dynam. Syst. 42 (2022), 3152-3166. [Lin99] E. Lindenstrauss, Mean dimension, small ...

work page 1977

[7] [7]

Lindenstrauss and B

[L W00] E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math. 115 (2000), 1-24. [L01] P. Liu, Dynamics of random transformations: smooth ergodic theory, Erg. Theory Dynam. Syst. 21 (2001), 1279-1319. [MT15] S. Matsuo and M. Tsukamoto, Brody curves and mean dimen- sion, J. Amer. Math. Soc. 28 (2015), 159-182. [Mat95] P. Mattila, The ge...

work page 2000

[8] [8]

[MYC19] X. Ma, J. Yang and E. Chen, Mean topological dimension for random bundle transformations, Erg. Theory Dynam. Syst. 39 (2019), 1020-1041. [MT19] T. Meyerovitch and M. Tsukamoto, Expansive multiparame- ter actions and mean dimension, Trans. Amer. Math. Soc. 371 (2019), 7275-7299. 27 [Mis75] M. Misiurewicz, A short proof of the variational principle ...

work page 2019

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Ruelle, Statistical mechanics on a compact set with Zν action satisfying expansiveness and speciﬁcation, Trans

[Rue73] D. Ruelle, Statistical mechanics on a compact set with Zν action satisfying expansiveness and speciﬁcation, Trans. Amer. Math. Soc. 187 (1973), 237-251. [Rud87] W. Rudin, Real and complex analysis, Third edition, McGraw- Hill Book Co., New York,

work page 1973

[10] [10]

xiv+416 pp. [S22] R. Shi, On variational principles for metric mean dimension, IEEE Trans. Inform. Theory 68 (2022), 4282-4288. [ST21] M. Shinoda and M. Tsukamoto, Symbolic dynamics in mean dimension theory, Erg. Theory Dynam. Syst. 41 (2021), 2542-

work page 2022

[11] [11]

[Tsu18] M

http://math.stanford.edu/ lms/ntu-gmt- text.pdf. [Tsu18] M. Tsukamoto, Mean dimension of the dynamical system of Brody curves, Invent. Math. 211 (2018), 935-968. [Tsu20] M. Tsukamoto, Double variational principle for mean dimen- sion with potential, Adv. Math. 361 (2020), 106935, 53 pp. [Tsu24] M. Tsukamoto, Rate distortion dimension of random Brody curve...

work page arXiv 2018

[12] [12]

[YCZ22] R. Yang, E. Chen and X. Zhou, On variational princi- ple for upper metric mean dimension with potential, arXiv: 2207.01901. [YCZ23] R. Yang, E. Chen and X. Zhou, Some notes on variational principle for metric mean dimension, IEEE Trans. Inform. The- ory 69 (2023), 2796-2800. 1.School of Mathematical Sciences and Institute of Mathema tics, Nanjing ...

work page arXiv 2023