Attracting cycles and post-critically finite maps

A theme which I hope to pursue in multiple posts on this blog is that even if one is interested mainly in complex dynamics, it is often useful to employ p-adic (or more generally non-Archimedean) methods.  There are a number of relatively recent research papers which illustrate and support this thesis.  The one I want to talk about today is this beautiful paper of Benedetto, Ingram, Jones, and Levy, henceforth referred to as [BIJL].

From a complex dynamicist’s point of view, the main result of [BIJL] is the following theorem:

Theorem 1: For any fixed integers d ≥ 2 and B ≥ 1, there are only finitely many conjugacy classes of post-critically finite rational maps of degree d which can be defined over a number field of degree at most B, except (when d is a perfect square) for the infinite family of flexible Lattès maps.

To explain the terms used in the statement, suppose f is a rational map of degree d with complex coefficients.  We say that f is post-critically finite (henceforth denoted PCF) if the orbit of the critical points of f under iteration is finite.  PCF maps play a fundamental role in complex dynamics, roughly speaking because many dynamical features of f can be read off from the behavior of the critical points under iteration.  One source of examples are the flexible Lattès maps, which can be defined whenever d=m2 is a perfect square: these consist of all degree m rational maps obtained as the map on x-coordinates induced by multiplication-by-m on some elliptic curve.  (McMullen calls such maps affine in the paper cited below.)

As an illustration of the theorem, the only quadratic polynomials of the form z2 + c with c a rational number which are PCF are z2, z2-1, and z2-2, and every quadratic polynomial is conjugate to a polynomial of this form. Continue reading