Algebra & Number Theory

Bifurcations, intersections, and heights

Laura DeMarco

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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.

Article information

Algebra Number Theory, Volume 10, Number 5 (2016), 1031-1056.

Received: 16 June 2015
Revised: 10 February 2016
Accepted: 10 March 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37P30: Height functions; Green functions; invariant measures [See also 11G50, 14G40]
Secondary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations 11G05: Elliptic curves over global fields [See also 14H52]

dynamics of rational maps canonical height stability


DeMarco, Laura. Bifurcations, intersections, and heights. Algebra Number Theory 10 (2016), no. 5, 1031--1056. doi:10.2140/ant.2016.10.1031.

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