WHATIS... ? the p-adic Mandelbrot Set? Joseph H. Silverman Before attempting to answer the title question, are complete, they are not algebraically closed; so we must first answer two preliminary questions: just as it is often better to work with the complete “What are the p-adic numbers?” and “What is the algebraically closed field of complex numbers C, (classical complex) Mandelbrot set?” We start with it is also often better to work with the field Cp, the former. the smallest complete algebraically closed field A standard characterization of the real num- containing Qp. But we note that Cp is a monster of bers R is as the smallest field containing Q that a field; it is not even locally compact! is complete with respect to the “usual” absolute The classical (degree 2 complex) Mandelbrot value jrj = maxfr; −rg on Q, where we recall 1 set M arises in studying the dynamics of the that a field is complete if every Cauchy sequence 1 simplest family of nonlinear functions, which is converges. But there are other absolute values the set of quadratic polynomials f (x) = x2 + on Q. Ostrowski showed that, up to a natural c equivalence, there is one absolute value for each c. Dynamicists study the effect of repeatedly prime number p. Writing a rational number r as applying the map fc to an initial point a 2 C; i.e., k a O = r = p with p not dividing ab, the p-adic absolute they study how the points in the orbit fc (a) b 2 n −k a; fc (a), f (a), : : : move around C, where f = fc ◦ value of r is defined by jrjp = p . Intuitively, two c c rational numbers r and s are p-adically close if the fc ◦· · ·◦fc denotes the nth iterate of fc . Of particular numerator of their difference is divisible by a large interest is the Julia set J(fc ) of fc , which is the set power of p. Then the field of p-adic numbers Qp of initial points where the iterates of fc behave is the smallest field containing Q that is complete chaotically. The Julia set may also be described as with respect to the p-adic absolute value j · jp. The the boundary of the set of initial points a 2 C whose p-adic numbers were invented by Hensel in the orbit Ofc (a) is bounded. Surprisingly, the geometry nineteenth century. They are analogous in many of J(fc ) is heavily influenced by the orbit of the ways to R. For example, the field Qp is locally single point 0. For example, a famous theorem compact for the topology induced by j · jp, and of Fatou and Julia (discovered independently) one can do p-adic analysis with p-adic power series. says that if Ofc (0) is bounded, then J(fc ) is A significant difference between R and Qp is that connected (although generally quite fractal-like), the p-adic absolute value satisfies the “ultrametric” and otherwise J(fc ) is totally disconnected. This triangle inequality jr + sjp ≤ max jrjp; jsjp . This dichotomy divides the parameter space of c implies that the unit disk x 2 Qp : jxjp ≤ 1 is a values into two pieces. The Mandelbrot set M1 compact subring of Q , which is nice, but it also p is the set of parameters c 2 C such that the implies that Qp is totally disconnected, which is orbit Ofc (0) is bounded, or equivalently, such not so nice. Also, although the fields R and Qp that J(fc ) is connected. You have undoubtedly Joseph H. Silverman is professor of mathematics at Brown seen pictures of the incredibly complicated and University. His email address is [email protected]. beautiful Mandelbrot set. It has become one of the DOI: http://dx.doi.org/10.1090/noti1038 most ubiquitous images in all of mathematics, and 1048 Notices of the AMS Volume 60, Number 8 the study of its geometry has occupied generations where of course we now use the p-adic absolute n of mathematicians. value to determine whether the orbits fc (γi) The modern theory of complex dynamics dates of the critical points are bounded. If p is large, from the fundamental work of Fatou and Julia then Mp;d is again boring, as shown by the around 1920. The study of p-adic dynamics is more following result that has long been “well known to recent, where initial investigations in the 1980s by the experts,” but seems to have first been written Herman, Yoccoz, and others led to an explosion of down in [1]. interest starting in the 1990s with groundbreaking Theorem 1. If p ≥ d, then M is a polydisk, results by Benedetto, Bézivin, Hsia, and Rivera- p;d d−1 Letelier. The classical Mandelbrot set is defined Mp;d = fc 2 Cp : jcijp ≤ 1 for all 1 ≤ i ≤ d − 1 : in terms of whether f n(0) is bounded or goes c The fact that M is a disk for all p, combined to infinity as n ! 1. We can use exactly the same p;2 with Theorem 1, seems to have discouraged people definition to define the p-adic Mandelbrot set Mp. from studying p-adic Mandelbrot sets, but recent Thus c 2 C is in M if and only if f n(0) is p p c p work by Anderson has shown that when p < d, bounded as n ! 1. The only change is that we’ve the p-adic Mandelbrot set Mp;d has a complicated replaced the usual absolute value on with the C geometric structure that may rival the geometry p-adic absolute value on . However, using the Cp of the classical complex Mandelbrot sets. ultrametric property of j · jp, it is very easy to see n that fc (0) p is bounded if and only if jcjp ≤ 1, so Example 2. Consider the action of the polynomial 3 3 3 Mp = c 2 Cp : jcjp ≤ 1 is the closed unit disk, g(x) = x − 4 x− 4 on C2. The critical points of g(x) 1 which really is not very interesting. are ± 2 , and they both have finite orbits, since If that were the end of the story, then we’d 1 g 1 1 g g have wasted a lot of ink, since the title question − ----------------------------! − and ----------------------------! −1 ----------------------------! −1: 2 2 2 would have a single word answer: boring. Luckily, − 3 − 3 2 M M matters become more interesting when we look at Hence ( 4 ; 4 ) 2;3, so 2;3 is not contained 3 Mandelbrot sets associated with polynomials of in the unit polydisk. (Note that j 4 j2 = 4 > 1.) higher degree. But first we ask why it is that the More generally, consider the one-parameter 2 orbit of the particular point 0 for fc (x) = x + c has 3 3 2 1 3 family of polynomials gt (x) = x − 4 t x− 4 (t +2t), such a profound influence on the dynamics of fc . so g(x) = g1(x). The critical points of gt (x) are the What makes the point 0 so important? The answer 1 1 fixed point γ1 = − 2 t and the point γ2 = 2 t, so gt is that 0 is the (unique) critical point of fc , i.e., the 1 0 is critically bounded if and only if the orbit of t derivative fc (x) vanishes at 0, and thus there is 2 is bounded. One can show that gt (x) is critically no neighborhood of 0 on which fc is one-to-one. We now fix d ≥ 2, and for each (d − 1)-tuple bounded for the sequence of parameter values 2k d−1 t = 1 + 2 converging 2-adically to 1, while it is c = (c1; : : : ; cd−1) 2 C , we define a (normalized) degree-d polynomial critically unbounded for the sequence of parameter values t = 1 + 3 · 22m+1, also converging 2-adically d d−2 d−3 fc (x) = x + c1x + c2x + · · · + cd−2x + cd−1: 3 3 to 1. Thus (− 4 ; − 4 ) is on the boundary of M2;3. (Every degree-d polynomial can be put into this Computations in [1] show that the geometry 3 3 form by conjugating by a linear polynomial. This of M2;3 \ Q2 in a 2-adic neighborhood of (− 4 ; − 4 ) conjugation does not materially affect the dy- is extremely complicated, and they further suggest namics.) The polynomial fc (x) has d − 1 critical that this neighborhood is geometrically equivalent points γ1; : : : ; γd−1 (counted with multiplicity) (in some not yet precisely defined sense) to a 1 whose orbits similarly have a profound influence 2-adic neighborhood of the critical point 2 in the on the dynamics of fc . We say that fc is criti- Julia set J(g1) \ Q2. This is our first inkling of cally bounded if the orbits Ofc (γi) of all of the a potential p-adic analogue of a famous classical critical points are bounded. Then the degree-d result over C that says that a Misiurewicz point c d−1 Mandelbrot set M1;d is the set of c 2 C whose in M1;2 has a neighborhood that is quasi-similar associated polynomials fc are critically bounded. to a neighborhood of the critical point 0 in the If we work over C, then these higher degree ana- complex Julia set J(fc ). logues of M1;2 are extremely complicated, with If p < d, then the p-adic Mandelbrot set Mp;d is higher-dimensional fractal-like boundaries.