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PreprintCC BY 4.0arXiv

Dynamical relations on ℙ1, higher bifurcations, and arithmetic bigness

  • Laura DeMarco
  • Niki Myrto Mavraki
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Abstract

In this article, we study algebraic relations among rational maps on ℙ1 and among their preperiodic points. Working over ℂ, we characterize algebraic families of positive-dimensional varieties in (ℙ1)N and tuples of maps (f1, …, fN) for which the subvariety is dynamically special for a Zariski-dense set of parameters. Moreover, outside of well-understood elliptic phenomena, we show that all families of maps and subvarieties that bifurcate are maximally bifurcating. Working over ℚ̅, we characterize arithmetic and geometric potential bigness for the canonical line bundle associated to families of endomorphisms of (ℙ1)N when restricted to the subvarieties. As consequences, we obtain uniform bounds for the periods and preperiods of relations among non-exceptional maps, and we show uniform bounds on the geometry of preperiodic points satisfying these relations. Over ℚ̅, our results yield corresponding uniform bounds for points and subvarieties satisfying a height gap.

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Cite

BibTeXRIS
BibTeX
@misc{demarco2026dynamical,
  title = {{Dynamical relations on $\mathbbP^1$, higher bifurcations, and arithmetic bigness}},
  author = {Laura DeMarco and Niki Myrto Mavraki},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40147},
  eprint = {2610.40147},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40147},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Dynamical relations on ℙ^1, higher bifurcations, and arithmetic bigness
AU  - Laura DeMarco
AU  - Niki Myrto Mavraki
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40147
UR  - https://arxiv.org/abs/2610.40147
AB  - In this article, we study algebraic relations among rational maps on ℙ^1 and among their preperiodic points. Working over ℂ, we characterize algebraic families of positive-dimensional varieties in (ℙ^1)^N and tuples of maps (f_1, …, f_N) for which the subvariety is dynamically special for a Zariski-dense set of parameters. Moreover, outside of well-understood elliptic phenomena, we show that all families of maps and subvarieties that bifurcate are maximally bifurcating. Working over ℚ̅, we characterize arithmetic and geometric potential bigness for the canonical line bundle associated to families of endomorphisms of (ℙ^1)^N when restricted to the subvarieties. As consequences, we obtain uniform bounds for the periods and preperiods of relations among non-exceptional maps, and we show uniform bounds on the geometry of preperiodic points satisfying these relations. Over ℚ̅, our results yield corresponding uniform bounds for points and subvarieties satisfying a height gap.
ER  -