Algebra & Number Theory
- Algebra Number Theory
- Volume 10, Number 5 (2016), 1031-1056.
Bifurcations, intersections, and heights
We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps , parameterized by 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.
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
Permanent link to this document
Digital Object Identifier
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]
DeMarco, Laura. Bifurcations, intersections, and heights. Algebra Number Theory 10 (2016), no. 5, 1031--1056. doi:10.2140/ant.2016.10.1031. https://projecteuclid.org/euclid.ant/1510842534