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arxiv: 2606.09031 · v1 · pith:4ZKLKXIA · submitted 2026-06-08 · math.DS · math.AG

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

Ethan Cohen This is my paper

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classification math.DS math.AG
keywords random skew productsmaximal entropy measurescomplex surfacesfiber entropyFurstenberg exponentlimit currentsrandom dynamicsautomorphisms
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The pith

A unique measure of maximal fiber entropy exists for skew products arising from random automorphisms on compact complex surfaces.

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

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.

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.

What carries the argument

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

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.

Where Pith is reading between the lines

These are 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.

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.

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.

read the original 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.

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 / 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)
  1. [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.
  2. 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.
  3. [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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 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.

axioms (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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discussion (0)

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