Vol. 10, No. 5, 2016

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Bifurcations, intersections, and heights

Laura DeMarco

Vol. 10 (2016), No. 5, 1031–1056
Abstract

We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps ft : 1() 1(), 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