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

Quadratic reciprocity and Zolotarev’s Lemma

I want to explain a beautiful proof of the Law of Quadratic Reciprocity from c. 1874 due to Egor Ivanovich Zolotarev (1847-1878). Some time ago I reformulated Zolotarev’s argument (as presented here) in terms of dealing cards and I posted a little note about it on my web page. After reading my write-up (which was unfortunately opaque in a couple of spots), Jerry Shurman was inspired to rework the argument and he came up with this elegant formulation which I think may be a “proof from the book”.  The following exposition is my own take on Jerry’s argument.  The proof requires some basic facts about permutations, all of which are proved in this handout.

Let m and n be odd relatively prime positive integers.  You have a stack of mn playing cards numbered 0 through mn-1 and you want to deal them onto the table in an m \times n rectangular array.  Consider the following three ways of doing this:

Row deal (\rho) : Deal the cards into rows, going left to right and top to bottom.

rho Continue reading