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Dimension of the equilibrium measure on Pn(C)\mathbb{P}^n(\mathbb{C})

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.CV — Complex Variables

Contributed by Snir Ben Ovadia, Aaron Brown

Version 2 / Oct 20, 2026 / CC BY 4.0

Abstract

Consider the dynamics of endomorphisms f:Pn(C)→Pn(C)f: \mathbb{P}^n(\mathbb{C})\to \mathbb{P}^n(\mathbb{C}) of degree d≥2d\geq 2. This note presents an explicit and elementary counterexample to the dimension formula for the equilibrium measure μ\mu conjectured by Binder--DeMarco, coming from the same family of quadratic skew products that appeared recently by Bianchi--He. In addition, the note shows that the one-sided inequality dimHμ≥(log⁡d)∑i=1nλi−1\mathrm{dim}_{\mathrm{H}}\mu\geq (\log d)\sum_{i=1}^n \lambda_i^{-1} holds for the equilibrium measure of every degree-dd holomorphic endomorphism of Pn\mathbb P^n. The proof adapts Dupont's volume-growth method to the unstable flag: combined with Gromov's orbit-graph argument, it bounds the cumulative partial entropies. Qian--Xie's extension of Ledrappier--Young theory then yields the inequality.

Provenance statement

Produced with light prompting of ChatGPT Astra, providing [C. Dupont, On the dimension of invariant measures of endomorphisms of CPk, Math. Ann. 349 (2011), no. 3, 509–528. doi:10.1007/s00208-010-0519-1]. We asked in what sense the two-dimensionality of the analysis there is essential. We were surprised by the produced manuscript, with its short proofs that extend the geometric arguments of Dupont, while giving an explicit counterexample to the Binder-DeMarco conjecture. Version 2 was obtained by suggesting additional references for proper attribution. We take no ownership of the proofs and only wish to share this result with the rest of the community.

References

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  1. J.-Y. Briend and J. Duval, Deux caractérisations de la mesure d'équilibre d'un endomorphisme de \(\PP^k(\CC)\) , Publications Mathématiques de l'IHÉS 93 (2001), 145--159. Theorem 2, p. 146. doi:10.1007/s10240-001-8190-4 (https://doi.org/10.1007/s10240-001-8190-4).DOI
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Version history

  1. v1Submitted by Snir Ben OvadiaInitial depositSupersededOct 20, 2026
  2. v2Submitted by Snir Ben OvadiaSubmitted as a new versionCurrentOct 25, 2026