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=m^{2} 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 z^{2} + c with c a rational number which are PCF are z^{2}, z^{2}-1, and z^{2}-2, and every quadratic polynomial is conjugate to a polynomial of this form. Continue reading