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Home , Mandelbrot set

WHATIS... ? the p-adic Mandelbrot ? 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 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 |r| = max{r, −r} on Q, where we recall ∞ set M arises in studying the dynamics of the that a field is complete if every Cauchy ∞ simplest family of nonlinear functions, which is converges. But there are other absolute values the set of quadratic f (x) = x2 + on Q. Ostrowski showed that, up to a natural c equivalence, there is one 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 ∈ C; i.e., k a O = r = p with p not dividing ab, the p-adic absolute they study how the points in the fc (a) b 2  n −k a, fc (a), f (a), . . . move around C, where f = fc ◦ value of r is defined by |r|p = 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 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 | · |p. The the boundary of the set of initial points a ∈ C whose

p-adic numbers were invented by Hensel in the orbit Ofc (a) is bounded. Surprisingly, the 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 compact for the induced by | · |p, and of Fatou and Julia (discovered independently) one can do p-adic analysis with p-adic power . says that if Ofc (0) is bounded, then J(fc ) is A significant difference between R and Qp is that connected (although generally quite -like), the p-adic absolute value satisfies the “ultrametric”  and otherwise J(fc ) is totally disconnected. This triangle inequality |r + s|p ≤ max |r|p, |s|p . This  dichotomy divides the of c implies that the unit x ∈ Qp : |x|p ≤ 1 is a values into two pieces. The Mandelbrot set M∞ compact subring of Q , which is nice, but it also p is the set of parameters c ∈ 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 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 = {c ∈ Cp : |ci|p ≤ 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 → ∞. 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 ∈ 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 → ∞. 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 | · |p, it is very easy to see n that fc (0) p is bounded if and only if |c|p ≤ 1, so Example 2. Consider the action of the  3 3 3 Mp = c ∈ Cp : |c|p ≤ 1 is the closed , 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 ∈ 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 | 4 |2 = 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) ∈ 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 c d−1 Mandelbrot set M∞,d is the set of c ∈ C whose in M∞,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 M∞,2 are extremely complicated, with If p < d, then the p-adic Mandelbrot set Mp,d is higher-dimensional fractal-like boundaries. (generally) not contained in the unit polydisk, so it is It’s clear how we should define the p-adic an interesting problem to compute or estimate the analogue of the degree-d Mandelbrot set; it is the p-adic radius of Mp,d. For this purpose, we define set the p-adic critical radius Rp,f of a polynomial f (x) d−1 to be the maximum of |γ −α|p as γ ranges over the Mp,d = {c ∈ C : fc (x) is p-adically critically p critical points of f (x) (roots of f 0(x)) and α ranges bounded}, over the fixed points of f (x) (roots of f (x) − x).

September 2013 Notices of the AMS 1049 Then the critical radius of Mp,d, denoted R(Mp,d), A beautiful and deep result of Jones [3] says that is the maximum of Rp,f for f ∈ Mp,d. Conjecture 3 is true if p ≡ 3 (mod 4) and that a slightly weaker statement with an alternative notion Example 3. It is an easy consequence of Theorem 1 of density is true for all p ≥ 3. Jones’s proof begins that R(M ) = 1 for p ≥ d. On the other hand, the p,d by using the function field Chebotarev density polynomial from Example 2 satisfies theorem to reduce the problem to properties of the 3 1 1 g0(x) = 3x2 − = 3(x − )(x + ) action of the Galois group of Fp(t)/Fp(t) on the 4 2 2 iterated preimage tree of 0. He next constructs a and stochastic process that encodes information about 3 1 g(x) − x = (x + 1)(x − )(x + ), the group action and shows that this process is a 2 2 martingale. Finally, additional information about so R = 2. Hence R(M ) ≥ 2. 2,g 2,3 the group action is combined with a martingale Theorem 2 (Anderson [1]). Let p be a prime satisfy- convergence theorem to complete the proof. 1 p/(d−1) ing 2 d < p < d. Then R(Mp,d) = p . Further, R(Mp,2p) = 1. Acknowledgment I would like to thank Jackie Anderson, Rob Bene- To prove the lower bound for R(p, d), it suffices detto, Liang-Chung Hsia, Rafe Jones, and the to exhibit a single polynomial in M . Explicitly, p,d referees for their helpful comments. one can show that the polynomial xd−p(x − β)p has critical radius pp/(d−1) for the carefully chosen value References β = [(−p/d)p(1 − p/d)d−p]−1/(d−1). The upper [1] J. Anderson, Bounds on the radius of the p-adic bound in Theorem 2 is more difficult and requires Mandelbrot set, 2012, arXiv:1207.2628. an elaborate calculation with Newton polygons. It’s [2] M. Baker and R. Rumely, Potential Theory and possible that the argument in [1] can be extended to Dynamics on the Berkovich Projective Line, Mathe- 1 1 matical Surveys and Monographs, vol. 59, American compute R(Mp,d) for 3 d < p < 2 d, but it appears Mathematical Society, Providence, RI, 2010. to be a difficult√ problem to evaluate R(Mp,d) when, say, p < d. [3] R. Jones, Iterated Galois towers, their associated mar- tingales, and the p-adic Mandelbrot set, Compos. Math. Returning now to the classical complex Man- 143 (5) (2007), 1108–1126. delbrot set M∞,2, we consider the collection H∞,2 [4] J. H. Silverman, The Arithmetic of Dynamical Systems, of hyperbolic maps, which is the set of parame- Chapter 5, Graduate Texts in Mathematics, vol. 241, ter values c ∈ M∞,2 such that the orbit of the Springer, , 2007. 2 critical point 0 of fc (x) = x + c converges to an attracting cycle. In other words, c ∈ H∞,2 if and n only if limn→∞ fc (0) converges to a point α ∈ C m m 0 satisfying fc (α) = α and (fc ) (α) < 1 for some m ≥ 1. It is known that H∞,2 is an open sub- set of M∞,2, and the Lebesgue measures satisfy 1.503 ≤ µ(H∞,2) ≤ µ(M∞,2) ≤ 1.562. The famous Hyperbolicity Conjecture asserts that H∞,2 equals the entire interior of M∞,2. The p-adic analogue Hp,2 is defined similarly; we simply replace C with Cp and use | · |p in place of the complex absolute value. Then, although Hp,2 is a subset of the (boring) closed unit disk Mp,2, it turns out that Hp,2 itself is quite complicated. A first reduction is to note that Hp,2 is the full inverse image under the “reduction mod p map” of the set  m H p,2 = c ∈ Fp : fc (0) = 0 in Fp for some m ≥ 1 .

(Here Fp denotes an algebraic closure of the finite field Fp.) It is conjectured that H p,2 is quite small, in contrast to the complex hyperbolic set H∞,2, which has positive . Conjecture 3. Let p ≥ 3. Then

#(H p,2 ∩ Fpk ) lim = 0. k→∞ pk

1050 Notices of the AMS Volume 60, Number 8