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The Binder-DeMarco-Dupont Inequality

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

Contributed by Snir Ben Ovadia ↗, Aaron Brown, Laura DeMarco

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

In the setting of endomorphisms of Pk(C)\mathbb{P}^k(\mathbb{C}), a dimension formula for the measure of maximal entropy was conjectured by Binder and DeMarco in 2003, asserting that $$\mathrm{dim}_\mathrm{Hausdorff}(\mu)=\log d\sum_{i=1}^k \frac{1}{\chi_i},$$ where dd is the degree of the (non-invertible) endomorphism and χi\chi_i's are the Lyapunov exponents with multiplicity. Indeed, this is well-defined as all Lyapunov exponents are known to be positive due to Briend and Duval. The most significant progress is due to Dupont, who in complex-dimension 2, proved the dimension lower-bound holds; And recently by Genes, who proved the equality for two-dimensional polynomial skew-product maps which satisfy a Lyapunov exponent condition. Recently, He and Bianchi were able to find a family of simple counter-examples. In Appendix A we present a single ``hands on'' description of such a counter example, which is given by a quadratic polynomial skew-product. In this manuscript, we provide the proof, which was achieved by lightly prompting ChatGPT, of extending Dupont’s inequality to all complex dimensions. We wish to share this manuscript with the community as we believe that this provides a full picture of the recently settled Binder-DeMarco conjecture. We do not claim ownership of the proof.

Provenance statement

Lightly prompted by 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] and asking in what sense the two-dimensionality of the analysis there is essential. After light prompting with ChatGPT Astra, it produced a manuscript which builds upon the geometric arguments of Dupont and extends them to higher dimensions.

References

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  2. T.-C. Dinh and N. Sibony, Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings , arXiv:0810.0811 (https://arxiv.org/abs/0810.0811).arXiv
  3. C. Dupont, On the dimension of invariant measures of endomorphisms of CPk\mathbb{CP}^k , Math. Ann. 349 (2011), no. 3, 509--528. doi:10.1007/s00208-010-0519-1 (https://doi.org/10.1007/s00208-010-0519-1). Author's manuscript (https://perso.univ-rennes1.fr/christophe.dupont/Articles/dimendo2.pdf).DOI
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  6. I. Binder and L. DeMarco, Dimension of pluriharmonic measure and polynomial endomorphisms of \(\CC^n\) , International Mathematics Research Notices 2003 , no. 11, 613--625. Conjecture 1.3, p. 615. Author's manuscript (https://people.math.harvard.edu/~demarco/IMRN.pdf).link
  7. 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
  8. J.-Y. Briend and J. Duval, Erratum to: Deux caractérisations de la mesure d'équilibre d'un endomorphisme de \(\PP^k(\CC)\) , Publications Mathématiques de l'IHÉS 109 (2009), 295--296. doi:10.1007/s10240-009-0020-0 (https://doi.org/10.1007/s10240-009-0020-0). The multiplicity lemma is replaced by a version for an iterate, sufficient for the subsequent arguments.DOI
  9. M. He and F. Bianchi, History-dependent unstable directions and the Binder--DeMarco conjecture , arXiv:2610.05602 (https://arxiv.org/abs/2610.05602).arXiv
  10. C. Genes, Dimension of the Green measure of skew products with dominant exponent in the fibers , arXiv:2610.19812 (https://arxiv.org/abs/2610.19812v1).arXiv

Version history

  1. v1Submitted by Snir Ben OvadiaInitial depositCurrentOct 20, 2026