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.

For polynomials, Theorem 1 was proved earlier in this paper of Patrick Ingram.  The polynomial case is much simpler than the case of general rational maps, for example because one does not have to worry about how to exclude the flexible Lattès maps (since they are never polynomials).  It is also simpler because there is a nice way to parametrize of the space of degree d polynomials in terms of their critical points and there is a corresponding easy-to-study and explicit ‘critical height function’ on this parameter space.

The proof of Theorem 1 relies on some deep results in both complex and p-adic dynamics.  I will attempt to explain the basic ideas behind the proof.

If z is a fixed point of f, the derivative of f at z is called the multiplier of z with respect to f.  More generally, if C is a cycle of period n, one can define the multiplier of C with respect to f as the multiplier of any point in C with respect to fⁿ (the n^th iterate of f).  This is well-defined by the chain rule.  A periodic cycle is called attracting if its multiplier has absolute value less than one.  The name comes from the fact that (by an elementary Taylor approximation argument) points near an attracting cycle will move closer and closer to the cycle under iteration.  An important classical theorem of Fatou asserts that every attracting cycle attracts a critical point of f.  In particular, f can have at most 2d-2 attracting cycles.  Fatou’s theorem also implies that a PCF map cannot have an attracting cycle with nonzero multiplier, since otherwise some critical point would be strictly attracted to the cycle and therefore have infinite orbit.

There is a beautiful and deep theorem of Curt McMullen (Corollary 2.3 in this paper) asserting that there is an integer N = N(d) such that, except for the flexible Lattès maps, the conjugacy class of a rational map of degree d is determined, up to finite number of choices, by the multipliers of all periodic cycles of length at most N.  (The infinitely many flexible Lattès maps of a given degree have the same multiplier spectrum, so it is necessary to exclude them in this theorem.)  So in order to prove Theorem 1, it suffices to prove that there are only finitely many choices for the multiplier λ of an n-cycle for f if n ≤ N.

To get the idea of how this can be proved, let’s consider the special case B=1.  In other words, f is a PCF map with rational coefficients.  We may assume without loss of generality that λ ≠ 0.

Claim: There is a constant M=M(d) such that all real or complex absolute values of λ^-1, as well as all p-adic absolute values of λ^-1, are bounded by M.

This implies that there are only finitely many choices for the (degree at most d^N) minimal polynomial of λ^-1, hence at most finitely many choices for λ^-1, hence only finitely many choices for λ.  The Claim is proved as a consequence of the other main theorem in [BIJL] (Theorem 2 below), which says that in the p-adic topology, every sufficiently attracting cycle attracts a critical point.

In order to make this precise, assume now that f is a rational map of degree d with coefficients in some p-adic field (which for the purposes of this post we take to mean a non-Archimedean field of characteristic zero and residue characteristic p).  As in the complex case, a periodic cycle is called attracting if its multiplier has absolute value less than one, and points near an attracting cycle will move closer and closer to the cycle under iteration.  It is no longer true in this setting that every attracting cycle attracts a critical point of f; for example, the map f(z)=z^p has infinitely many periodic cycles, all of which are attracting, but the only critical points of f are 0 and ∞.  So the naïve p-adic analogue of Fatou’s theorem is false.  Nevertheless, the following holds:

Theorem 2: There is a constant ε = ε(p,d)>0 such that if z is a fixed point of f whose multiplier λ satisfies 0 < |λ|_p < ε, then there is a critical point of f which is strictly attracted to z.  Moreover, if p>d then one can take ε(p,d) = 1.

This easily implies the Claim, and hence Theorem 1, in the special case B=1 by applying Theorem 2 to each fⁿ with 1 ≤ n ≤ N.  The general case of Theorem 1 follows from the theory of Weil heights, since the above argument shows that h(λ) is bounded by a constant depending only on d and by Northcott’s Theorem (see for example Theorem 1.7 here) there are only finitely many algebraic numbers λ with bounded height and bounded degree.

The proof of Theorem 2 takes place (as every good proof should) on the Berkovich projective line.  I will blog about this beautiful space (and the proof of Theorem 2) some other time; for now, suffice it to say that it is the “correct” place to do p-adic analysis and the authors of [BIJL] give a very nice summary of all the facts they need in Section 3.  It should be noted that one does not have to use Berkovich spaces here but it simplifies the argument substantially.  To wit, in verson 1 of [BIJL] on the arXiv, Theorem 2 is proved in a Berkovich-free way via a rather harrowing Newton polygon argument which occupies more than 10 pages; in version 2, the proof is reduced to just over 2 pages and it becomes much more conceptual in the process.

Concluding observations:

1. In Section 4 of Ingram’s paper referenced above, there is an explicit list of seven PCF polynomials such that every PCF cubic polynomial defined over the rational numbers is conjugate to one of the polynomials on the list.  In [BIJL] it is proved that if f is a PCF degree 2 rational map and λ is the multiplier of a fixed point of f, then h(λ) ≤ log 4.  In principle, this is enough to give an explicitly computable finite set of possibilities for all PCF quadratic rational maps defined over the field of rational numbers, but the authors write that this is a “daunting computation”.  [Note added: The daunting computation has now been done, see Patrick Ingram’s comment below. Also, as pointed out by MathHound in the comments, there are in fact 15 cubic PCF polynomials up to conjugacy; they are determined in this paper.]

2. The constant ε = ε(p,d) in Theorem 2 can be taken to be the d^th power of the minimum of |m|_p over all positive integers m from 1 up to d.  For polynomial maps one can instead take the m^th power of the minimum, and this estimate is best possible.

3. One can deduce from Corollary 1.7 in [BIJL] that for any integer n and any field k of characteristic different from 2, there are at most finitely many isomorphism classes of quadratic rational maps having a finite postcritical orbit of size n.  A completely different proof of this fact was recently given in this arXiv preprint by Richard Pink.  If we take k to be the field of complex numbers, this is a special case of Thurston’s Rigidity Theorem, which plays a key role in the proof of McMullen’s theorem mentioned above.

4. From a number theorist’s point of view, there are interesting analogies between PCF maps and elliptic curves with complex multiplication.  For example, the “arboreal Galois representation” associated to a rational map f defined over a number field K will have much smaller image than one expects in general if f is post-critically finite, just as the l-adic Galois representation associated to an elliptic curve over K having CM is much smaller than for non-CM elliptic curves.  The locus of (non-Lattes) PCF maps forms a countable dense set of “special” algebraic points in the moduli space of rational maps of degree d, just as the CM elliptic curves give a countable dense set of “special” algebraic points in the moduli space of elliptic curves.  From this point of view, Theorem 1 can be viewed as an analogue of the classical fact that for any positive integer B there are only finitely many CM elliptic curves defined over a number field of degree at most B.  (This is a consequence of class field theory, the theory of complex multiplication, and Gauss’s “Class Number Conjecture”, proved by Heilbronn in 1934.)