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arxiv: 2604.14964 · v1 · submitted 2026-04-16 · 🧮 math.DS

Induced and nonlinear topological pressure for random dynamical systems

Cunyi Nan This is my paper

Pith reviewed 2026-05-10 10:00 UTC · model grok-4.3

classification 🧮 math.DS
keywords induced topological pressurenonlinear fiber pressurerandom dynamical systemsfiber topological pressurevariational principlespanning setsseparated sets
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The pith

Induced fiber pressure equals the pseudo-inverse of classical fiber topological pressure in random dynamical systems.

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

The paper defines a non-averaged induced fiber pressure for random dynamical systems using spanning and separated sets. It proves this quantity is the pseudo-inverse of the classical fiber topological pressure and satisfies a corresponding variational principle. A separate nonlinear fiber pressure is introduced along with its own variational principles. The entire framework is extended to higher-dimensional cases. A sympathetic reader would care because these constructions remove the need for averaging over randomness while preserving the core thermodynamic formalism.

Core claim

We define a non-averaged induced fiber pressure via spanning and separated sets, characterize it as the pseudo-inverse of the classical fiber topological pressure studied previously, and establish the corresponding variational principle. We also define the nonlinear fiber pressure and prove the associated variational principles. Finally, we extend the combined theory to the higher-dimensional setting.

What carries the argument

The non-averaged induced fiber pressure, defined directly through spanning and separated sets and shown to act as the pseudo-inverse of the classical fiber topological pressure.

If this is right

  • Variational principles apply to the induced pressure without averaging.
  • Nonlinear fiber pressure admits its own variational principles in random systems.
  • The results carry over to higher-dimensional random dynamical systems.

Where Pith is reading between the lines

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

  • The non-averaged definition could simplify pressure calculations in systems where randomness makes averaging difficult or undefined.
  • This framework may extend naturally to other non-stationary or non-autonomous dynamical systems beyond the random case.

Load-bearing premise

The classical fiber topological pressure is well-defined for the random dynamical systems considered, and the pseudo-inverse characterization plus variational principles hold under the spanning/separated set definitions without additional hidden regularity conditions on the randomness.

What would settle it

A concrete random dynamical system where the induced pressure computed from spanning or separated sets does not numerically equal the pseudo-inverse of the classical fiber pressure.

read the original abstract

In this paper, we investigate induced and nonlinear fiber topological pressure for random dynamical systems. We define a non-averaged induced fiber pressure via spanning and separated sets, characterize it as the pseudo-inverse of the classical fiber topological pressure studied previously, and establish the corresponding variational principle. We also define the nonlinear fiber pressure and prove the associated variational principles. Finally, we extend the combined theory to the higher-dimensional setting.

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.

Referee Report

0 major / 2 minor

Summary. The paper defines a non-averaged induced fiber topological pressure for random dynamical systems using spanning and separated sets, characterizes this quantity as the pseudo-inverse of the classical fiber topological pressure from prior work, and proves the associated variational principle. It further introduces a nonlinear fiber pressure, establishes its variational principles, and extends the combined framework to higher-dimensional random dynamical systems.

Significance. If the definitions and proofs hold, the work extends the theory of topological pressure to induced and nonlinear settings in the random case, providing new variational principles that align with classical approaches via spanning/separated sets. This could offer additional tools for analyzing random dynamical systems in ergodic theory, particularly where induced maps or nonlinear functionals arise.

minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a brief explicit statement of the standing assumptions on the base transformation, fiber maps, and probability space that ensure the classical fiber pressure is well-defined before introducing the new quantities.
  2. [Introduction] Notation for the pseudo-inverse operation and the distinction between averaged and non-averaged pressures should be clarified with a short comparison table or remark to avoid potential confusion with related concepts in the literature on random pressures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation for minor revision. The referee's description accurately reflects the paper's contributions on non-averaged induced fiber topological pressure, its characterization via pseudo-inverse, the variational principle, the nonlinear fiber pressure, and the extension to higher dimensions. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the non-averaged induced fiber pressure directly via spanning and separated sets on the random dynamical system, then proves its characterization as the pseudo-inverse of the classical fiber pressure (assumed well-defined from prior setup) and the associated variational principle. The nonlinear fiber pressure is likewise introduced via an independent definition with its own variational principle. These steps rely on the given RDS setup (base transformation, fiber maps, probability space) without reducing any new quantity to a fitted parameter, self-referential equation, or unverified self-citation chain within the present work. The relation to prior classical pressure is a derived property rather than a definitional input.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background from ergodic theory and prior results on classical fiber topological pressure; no free parameters or invented physical entities are indicated in the abstract.

axioms (2)
  • domain assumption Classical fiber topological pressure is well-defined and admits a variational principle for the random dynamical systems under study.
    Invoked when characterizing the induced version as its pseudo-inverse and when extending variational principles.
  • domain assumption Spanning and separated sets can be used to define pressures in the fiber/random setting without averaging.
    Central to the definition of the non-averaged induced pressure.

pith-pipeline@v0.9.0 · 5341 in / 1377 out tokens · 34882 ms · 2026-05-10T10:00:50.751708+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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    [AKM65] R. L. Adler, G. Konheim, and M. H. McAndrew. Topological entropy.Transactions of the American Mathematical Society, 114(2):309–319, 1965. [BH22a] L. Barreira and C. Holanda. Higher-dimensional nonlinear thermodynamic formalism.Journal of Statis- tical Physics, 187(2), 2022. [BH22b] L. Barreira and C. Holanda. Nonlinear thermodynamic formalism for ...

    work page 1965

  2. [2]

    De Gruyter, 2022

    GDMSs, Lasota-Yorke Maps and Fractal Geometry. De Gruyter, 2022. [Wal75] P. Walters. A variational principle for the pressure of continuous transformations.American Journal of Mathematics, 97(4):937–971, 1975. [Wal82] P. Walters.An Introduction to Ergodic Theory. Springer, 1982. [WZZ25] X. Wang, Z. Zhang, and Y. Zhu. On entropy, pressure and variational p...

    work page 2022