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PreprintCC BY 4.0arXiv

History-dependent unstable directions and the Binder--DeMarco conjecture

  • Fabrizio Bianchi
  • Yan Mary He
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Abstract

We establish dimension bounds and an exact dimension formula for equilibrium measures of degree d holomorphic endomorphisms of ℙ2=ℙ2(ℂ) satisfying suitable expansion and domination conditions. Write λ1>λ2>0 for their Lyapunov exponents. If two inverse histories ending at the same point determine different fast directions, we prove that dimH μF>log d/λ1+log d/λ2 when λ2>log d, and that dimH μF=2log d/λ2 when λ2≥2log d. Applying these results to an explicit quadratic family, we disprove the Binder--DeMarco conjecture on a non-empty open set of holomorphic endomorphisms. The proof adapts the projection growth strategy of Li--Pan--Tong--Xu to the holomorphic setting, combining strong leaf geometry and the Ledrappier--Young theory for endomorphisms with a multidimensional extension of Wu's restricted sum estimate.

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Cite

BibTeXRIS
BibTeX
@misc{bianchi2026historydependent,
  title = {{History-dependent unstable directions and the Binder--DeMarco conjecture}},
  author = {Fabrizio Bianchi and Yan Mary He},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05602},
  eprint = {2610.05602},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05602},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - History-dependent unstable directions and the Binder--DeMarco conjecture
AU  - Fabrizio Bianchi
AU  - Yan Mary He
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05602
UR  - https://arxiv.org/abs/2610.05602
AB  - We establish dimension bounds and an exact dimension formula for equilibrium measures of degree d holomorphic endomorphisms of ℙ^2=ℙ^2(ℂ) satisfying suitable expansion and domination conditions. Write λ_1>λ_2>0 for their Lyapunov exponents. If two inverse histories ending at the same point determine different fast directions, we prove that dim_H μ_F>log d/λ_1+log d/λ_2 when λ_2>log d, and that dim_H μ_F=2log d/λ_2 when λ_2≥2log d. Applying these results to an explicit quadratic family, we disprove the Binder--DeMarco conjecture on a non-empty open set of holomorphic endomorphisms. The proof adapts the projection growth strategy of Li--Pan--Tong--Xu to the holomorphic setting, combining strong leaf geometry and the Ledrappier--Young theory for endomorphisms with a multidimensional extension of Wu's restricted sum estimate.
ER  -