The measure of maximal entropy for random skew products on compact complex surfaces

Ethan Cohen

The measure of maximal entropy for random skew products on compact complex surfaces

T0 review · 0 major / 3 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read A unique measure of maximal fiber entropy exists for skew products arising from random automorphisms on compact complex surfaces.

desk verdict Cohen gives a canonical maximal fiber entropy measure for random skew products on complex surfaces and equates its entropy to the Furstenberg exponent. read the letter →

arxiv 2606.09031 v2 pith:4ZKLKXIA submitted 2026-06-08 math.DS math.AG

Ethan Cohen This is my paper · ORCID

classification math.DSmath.AG

keywords randomskewproductsmaximalentropymeasurescomplexsurfacesfiberFurstenbergexponentlimitcurrentsdynamicsautomorphisms

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

The paper proves that when a Borel probability measure on the automorphism group of a compact complex surface satisfies logarithmic integrability and its support generates a non-elementary subgroup, the associated skew product has a unique invariant measure maximizing fiber entropy. This measure is constructed canonically from random limit currents, and its entropy value equals the Furstenberg exponent of the induced random action on cohomology. Under an added exponential moment condition the measure is also mixing. A sympathetic reader would care because the result supplies a concrete, computable object for the long-term statistical behavior of iterated random holomorphic maps on surfaces.

What carries the argument

Random limit currents, which canonically construct the measure of maximal fiber entropy and equate its entropy to the Furstenberg exponent.

What would settle it

An explicit measure μ whose support generates a non-elementary subgroup but for which either multiple distinct measures achieve the same fiber entropy or the entropy value differs from the Furstenberg exponent.

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Extended reading notes

Core claim

Let X be a compact complex surface. The skew product associated to a Borel probability measure μ on Aut(X) admits a unique invariant measure of maximal fiber entropy, assuming that μ satisfies a logarithmic integrability condition and that supp(μ) generates a non-elementary subgroup of Aut(X). We describe this measure canonically in terms of the random limit currents constructed by Cantat and Dujardin, and show that its fiber entropy is equal to the Furstenberg exponent of the associated random action on cohomology. Under an exponential moment assumption, we prove that it is mixing.

Load-bearing premise

That the support of μ generates a non-elementary subgroup of Aut(X), which is needed to produce well-defined non-degenerate limit currents and guarantee uniqueness.

Editorial extensions

If this is right

The fiber entropy of the unique maximal measure equals the Furstenberg exponent of the random cohomology action.
The measure is mixing whenever μ also satisfies an exponential moment condition.
The construction applies to every Borel probability measure on Aut(X) meeting the stated integrability and generation hypotheses.
Uniqueness of the maximal fiber entropy measure follows directly from the non-elementary generation assumption.

Reading between the lines

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

The same construction could be tested on other surfaces where analogous random limit currents have been shown to exist.
Explicit computation of the Furstenberg exponent for concrete finitely generated subgroups might now yield numerical values for maximal fiber entropies.
The mixing property under exponential moments suggests that correlation decay estimates could be derived from the same current-based description.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The core result is that for a measure μ on Aut(X) with the usual log-integrability and non-elementary support conditions, the skew product has a unique invariant measure of maximal fiber entropy. It is described explicitly via the Cantat-Dujardin random limit currents, and its fiber entropy equals the Furstenberg exponent of the induced action on cohomology. Under an exponential moment assumption the measure is mixing.

This is a direct extension of the deterministic picture to the random setting. The canonical description via the currents is the part that feels most useful; it turns an existence statement into something one can work with. The equality to the Furstenberg exponent is the cleanest link if the derivation goes through.

The hypotheses are the standard ones already used to get positive exponents and non-degenerate currents, so they do not look like an artificial restriction. The mixing statement under exponential moments is a natural strengthening.

The main limitation visible from the abstract is that the actual proofs are not here to check. The equality and uniqueness steps are the load-bearing parts, and without seeing how the currents are used to construct the measure or how the entropy computation avoids circularity, it is hard to gauge how much new work is required. No obvious gaps in the outline, but that is all one can say.

This is for people already working in random complex dynamics or ergodic theory on surfaces. A reader who knows the Cantat-Dujardin currents and Furstenberg exponents will get the most out of it. It is worth sending to referees; the statement is precise enough and the objects are well-motivated that a careful check could confirm a solid incremental result.

Referee Report

0 major / 3 minor

Summary. The paper proves that for a compact complex surface X and Borel probability measure μ on Aut(X) satisfying a logarithmic integrability condition with supp(μ) generating a non-elementary subgroup, the associated skew product admits a unique invariant measure of maximal fiber entropy. This measure is described canonically via the random limit currents of Cantat and Dujardin, its fiber entropy equals the Furstenberg exponent of the random action on cohomology, and the measure is mixing under an additional exponential moment assumption.

Significance. If the result holds, it canonically identifies the measure of maximal fiber entropy for these random skew products and equates its entropy to the Furstenberg exponent, providing a concrete link between random dynamics, limit currents, and cohomology actions on complex surfaces. The use of standard hypotheses to guarantee non-degenerate currents and the equality itself are strengths that connect entropy directly to the linear action without fitted parameters.

minor comments (3)

[Introduction] The introduction should include an explicit numbered statement of the main theorem (existence, uniqueness, canonical description, and entropy equality) with a forward reference to the section containing the proof.
Clarify the precise definition of 'fiber entropy' for the skew product (e.g., via the disintegration over the base or as an integral of conditional entropies) and confirm it matches the quantity whose maximality is asserted.
[Abstract] Add the full bibliographic reference for the Cantat–Dujardin random limit currents at first mention, rather than leaving it implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper establishes existence and uniqueness of a maximal fiber entropy measure for random skew products on complex surfaces, under logarithmic integrability and non-elementary generation assumptions. It canonically describes the measure via Cantat-Dujardin random limit currents and proves equality of fiber entropy to the Furstenberg exponent. These steps are presented as theorems derived from the given hypotheses and external constructions, not as self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The equality is a derived relation between independently defined quantities (fiber entropy of the invariant measure and the cohomology action exponent), with no reduction by construction visible from the abstract or stated claims. The assumptions are standard in the field and do not create circularity.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the logarithmic integrability condition and the non-elementary generation assumption, plus the prior existence of random limit currents from Cantat and Dujardin. No free parameters or invented entities are introduced in the abstract.

assumptions (2)

domain assumption Logarithmic integrability condition on μ
Required for the construction and uniqueness of the maximal entropy measure.
domain assumption supp(μ) generates a non-elementary subgroup of Aut(X)
Required to ensure the random action is sufficiently mixing and the limit currents are well-behaved.

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Cite this review

Pith. "Pith review of The measure of maximal entropy for random skew products on compact complex surfaces." pith.science (2026). https://pith.science/paper/4ZKLKXIA

@misc{pith2026260609031,
  author       = {Pith},
  title        = {Pith review of: The measure of maximal entropy for random skew products on compact complex surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZKLKXIA}},
  note         = {Machine review of arXiv:2606.09031}
}

abstract

Let $X$ be a compact complex surface. We prove that the skew product associated to a Borel probability measure $\mu$ on $\operatorname{Aut}(X)$ admits a unique invariant measure of maximal fiber entropy, assuming that $\mu$ satisfies a logarithmic integrability condition and that $\operatorname{supp}(\mu)$ generates a non-elementary subgroup of $\operatorname{Aut}(X)$. We describe this measure canonically in terms of the random limit currents constructed by Cantat and Dujardin, and show that its fiber entropy is equal to the Furstenberg exponent of the associated random action on cohomology. Under an exponential moment assumption, we prove that it is mixing.

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Reviewed June 27, 2026 · model on record in the stance chip above.