Bifurcations, intersections, and heights

DeMarco, Laura

doi:10.2140/ant.2016.10.1031

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Bifurcations, intersections, and heights

Laura DeMarco

Vol. 10 (2016), No. 5, 1031–1056

DOI: 10.2140/ant.2016.10.1031

Abstract

We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps $f_{t} : ℙ^{1} (ℂ) \to ℙ^{1} (ℂ)$ , 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.

Keywords

dynamics of rational maps, canonical height, stability

Mathematical Subject Classification 2010

Primary: 37P30

Secondary: 37F45, 11G05

Milestones

Received: 16 June 2015

Revised: 10 February 2016

Accepted: 10 March 2016

Published: 28 July 2016

Authors

Laura DeMarco
Department of Mathematics Northwestern University 2033 Sheridan Road Evanston, IL 60208-2730 United States

Vol. 10, No. 5, 2016