Bifurcations, intersections, and heights

Laura DeMarco

doi:10.2140/ant.2016.10.1031

Bifurcations, intersections, and heights

Laura DeMarco^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps f_t:ℙ¹(ℂ) → ℙ¹(ℂ), parameterized by t in a quasiprojective complex variety. We use this to prove one implication in the if-and-only-if statement of a certain conjecture on unlikely intersections in the moduli space of rational maps (see “Special curves and postcritically finite polynomials”, Forum Math. Pi 1 (2013), e3). We present the conjecture here in a more general form.

Original language	English (US)
Pages (from-to)	1031-1056
Number of pages	26
Journal	Algebra and Number Theory
Volume	10
Issue number	5
DOIs	https://doi.org/10.2140/ant.2016.10.1031
State	Published - 2016

Funding

The research was supported by the National Science Foundation, DMS-1517080.

Keywords

Canonical height
Dynamics of rational maps
Stability

ASJC Scopus subject areas

Algebra and Number Theory

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10.2140/ant.2016.10.1031

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AB - We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps ft:ℙ1(ℂ) → ℙ1(ℂ), parameterized by t in a quasiprojective complex variety. We use this to prove one implication in the if-and-only-if statement of a certain conjecture on unlikely intersections in the moduli space of rational maps (see “Special curves and postcritically finite polynomials”, Forum Math. Pi 1 (2013), e3). We present the conjecture here in a more general form.

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Bifurcations, intersections, and heights

Abstract

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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