Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps

Abstract

A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatou's classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.

No significant mathematical changes, but some (minor) changes in presentation

References in corpus (2)

• A census of quadratic post-critically finite rational maps defined over Q
• Multiplier Spectra and the Moduli Space of Degree 3 Morphisms on P1

Cited by in corpus (11)

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• Rigidity and height bounds for certain post-critically finite endomorphisms of projective space
• The McMullen Map in Positive Characteristic
• Minimally critical regular endomorphisms of A^N
• Minimally critical endomorphisms of P^N
• New Normal Forms For Degree Three Polynomials and Rational Functions
• Activity measures of dynamical systems over non-archimedean fields
• The critical orbit structure of quadratic polynomials in ${\mathbb{Z}}_p$