Attracting cycles in padic dynamics and height bounds for postcritically finite maps
Abstract
A rational function of degree at least two with coefficients in an algebraically closed field is postcritically 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 wellunderstood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of nonLattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a nonarchimedean version of Fatou's classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.
 Publication:

arXiv eprints
 Pub Date:
 January 2012
 arXiv:
 arXiv:1201.1605
 Bibcode:
 2012arXiv1201.1605B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Dynamical Systems
 EPrint:
 No significant mathematical changes, but some (minor) changes in presentation