Introduction
This thesis has four main elements: dynamics, algebraic number theory (in function fields), grouptheory, and probability. The dynamics provides inspiration for the problem, the algebraic number theory allows an important reformulation, the group theory does the heavy lifting, and theprobability theory synthesizes the group theory into a form that solves the problem. Chapter 1gives the dynamical background, Chapter 2 the number-theoretic translation, and Chapter 3 thegroup-theoretic results. Chapter 4 deals almost solely with probability, while Chapter 5 applies theprobability to give a solution to the problem.
In this introduction, we sketch of the origins of the problem, state the problem, and give a
detailed outline of our solution. The origins of the problem and the first stages of the solution lie
in the field of algebraic dynamics, which can be broadly defined as the study of function iteration
over algebraic/arithmetical sets, such as algebraic number fields and rings, polynomial rings, finitefields, p-adic fields, algebraic curves, etc. This relatively new field evolved naturally from the fieldof complex dynamics, whose roots stretch back to the work of Julia and Fatou in the early 1900s.
Complex dynamics enjoyed an explosion of deep results in the 1980s; one could note in particular
Benoit Mandelbrot’s popularization of the remarkable set that bears his name, Dennis Sullivan’s
No Wandering Domains theorem [36], and Curt McMullen’s proof of the nonexistence of generally
convergent algorithms in degree larger than three [23].
In the 1990s, some attention began to turn to p-adic analogues of the blossoming complex theory.
Michael Herman and Fields medalist Jean-Christophe Yoccoz [17] had already gone in this direction
with their 1983 paper on a non-Archimedean version of Siegel’s linearization theorem [34]. Rob
Benedetto proved a partial analogue of the No Wandering Domains theorem in 2000 [6] and Juan
Rivera-Letelier, a student of Yoccoz, fleshed out much of the theory of p-adic Fatou and Julia setsin the early part of the 2000s [29, 30]. An unexpected impetus for exploring p-adic dynamics hascome from physicists, who began to explore what the world looked like through p-adic eyes. The
first paper whose title contained the phrase “p-adic dynamics” appeared in the Physics literature in
1989 [37].
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The work presented in this thesis stems from a question regarding the degree to which p-adic
dynamics and complex dynamics are analogous. This question deals with the p-adic analogue of thecomplex Mandelbrot set. The complex Mandelbrot set, defined to be
M = {c ∈ C : 0 has a bounded orbit under iteration of z 2 + c}
not only created a stir among researchers, but its striking images reached an audience far beyond themathematical community. It has assumed a place as one of the most widely recognized mathematicalobjects. Thus the p-adic analogue of M , at first blush, seems to promise a similar treasure trove ofcomplexity. This promise unfortunately proves false, as the analogous set is simply the closed unit
disk (for p "= 2).

Peering into the set more closely, however, reveals a particular subset that is more interesting.

It is the subset of parameter values c such that z 2 + c is hyperbolic; see below for a definition ofhyperbolicity and see (1) for a definition of the set. Its complex analogue has been much studied[1, 14, 19, 22]. This complex analogue is a large subset of M , accounting for at least 96% of the area[11], and is conjectured to be the interior of M [24]. The problem this thesis sets out to resolve isto determine the “size” of the hyperbolic subset of the p-adic Mandelbrot set. We show that it is ina certain sense a measure zero subset, contrasting sharply with the complex case.
The first hurdle is to say what is meant by size: there is no suitable notion of measure on thep-adic analogue of C because it is not locally compact. The way around this, and the first step inthe solution of the above problem, is to use the reduction homomorphism to translate the problem
into one of dynamics over Fp . In Fp we define density measures closely related to the well-known
Dirichlet density and natural density (see e.g. [20]). The proof then follows a path through algebraic
number theory, then the Galois theory of function fields, and eventually into the realm of stochastic
processes, where it reaches its conclusion. This method of proof appears to be highly unusual, andmay be fruitful in answering other density questions regarding dynamically defined sets. We now
give a detailed outline of the argument.
In Chapter 1 we begin with some definitions and background. We call a rational function hyperbolic if all its critical points are attracted to attracting cycles (see page 8 for definitions of theseterms). Hyperbolic maps have many nice properties, and are the subject of the biggest unsolved
conjecture in complex dynamics (see page 9 for a statement and [24] for more detail). The map
z 2 + c has critical points at infinity and 0, and infinity is an attracting fixed point. Therefore z 2 + c
is hyperbolic if and only if 0 is attracted to an attracting cycle. Thus the set
H(C) = {c ∈ M : 0 is attracted to an attracting cycle of z 2 + c}

(1)

is the hyperbolic subset of M . We examine the analogous set defined over the field Cp , which is the3smallest complete, algebraically closed extension of Qp ; it is therefore the p-adic analogue of C. Let
Mp be the natural analogue of M over Cp . It’s easily seen that Mp = {c ∈ Cp : |c| ≤ 1} provided
that p "= 2. (Proposition 1.2). Throughout this discussion, we take p to be a prime different from 2.
However, the subset

H(Cp ) = {c ∈ Mp : 0 is attracted to an attracting cycle of z 2 + c}
is not so easily characterized. Letting φ : {|c| ≤ 1} → Fp be the reduction homomorphism (see(1.2)), we establish in Corollary 1.6 that H(Cp ) = φ−1 (H(Fp )), whereH(Fp ) = {α ∈ Fp : 0 is periodic under iteration of x2 + α}.
(Note that by periodic we mean that the orbit of 0 is a cycle; some authors refer to this as purelyperiodic.) We define two notions of density for subsets of Fp , called Dirichlet density and naturaldensity, that are closely related to the densities of the same names defined for subsets of primes
in Fp [x]. We denote Dirichlet density by δ and natural density by D (see (1.6) and (1.7) for thedefinitions). Our main result (Theorem 1.7) is that δ(H(Fp )) = 0, and its proof is the principal goalof all of our subsequent work. We also establish, using a similar method, a companion result: weshow D(H(Fp )) = 0 for p ≡ 3 mod 4, and we conjecture that D(H(Fp )) = 0 holds for all p "= 2.

In Chapter 2, we take the first steps toward a proof by giving two translations of the problem.

Although the definition of H(Fp ) says that the forward orbit of 0 under iteration of x2 + α is a cycle,

we focus in Section 2.1 on the inverse orbit of 0 under x2 + α. Clearly for any α ∈ Fp the forward
orbit of 0 under iteration of x2 + α is contained in Fp (α). If 0 is periodic, however, this forward
orbit coincides with one branch of the inverse orbit of 0 in Fp , and thus 0 has nth preimages inFp (α) for each n ≥ 1. We show (Proposition 2.1) that the converse is also true. Therefore, setting
fα = x2 + α, and denoting by fα−n (0) the set of nth preimages of 0 (in Fp ) under iteration of fα , wehave

H(Fp ) = {α ∈ Fp : fα−n (0) ∩ Fp (α) "= ∅ for all n ≥ 1}.

We can therefore define a sequence of sets
In = {α ∈ Fp : fα−n (0) ∩ Fp (α) "= ∅}(2)that serve as progressively better “approximations” of H(Fp ) in the sense that In ⊇ In+1 and!
H(Fp ) = n In . We show that if δ(In ) exists for all n and lim δ(In ) = 0, then δ(H(Fp )) = 0.
n→∞

In Section 2.2, we prove that δ(In ) exists and give a method of computing it using the Galois

groups of certain algebraic extensions of Fp (x). Our main tool in this endeavor is the Tchebotarev
Density theorem for function fields. In order to use it, we find a set of primes in Fp [x] that is4expressible in terms of the Artin symbol (see page 25 for a definition), and whose Dirichlet density(in the sense of (2.3)) is equal to δ(In ). Note that the set fα−n (0) consists of the roots of fαn , andthus fα−n (0) ∩ Fp (α) "= ∅ if and only if the factorization of fαn over Fp (α) contains at least one linear
term. There is a thus a close relationship between In and the following set of primes in Fp [x]:In = {p ⊆ Fp [x] : fxn has a linear factor mod p},
where fx = y 2 + x ∈ Fp (x)[y]. Indeed, we show that δ(In ) = δ(In ), where this second Dirichlet

density is the usual one for sets of primes in a function field (2.3). We then use some standard
arguments in algebraic number theory to show that In differs by only finitely many primes from a
set of primes defined in terms of the Artin symbol (2.14). This allows us to apply the Tchebotarev
Density theorem. Let Kn be the splitting field of fxn over K = Fp (x), and Gn = Gal (Kn /K). InTheorem 2.18 we show that δ(In ) exists for all n and equals
1# {g ∈ Gn : g fixes at least one root of fxn } .
#Gn(3)The same statement hold for D(In ), provided that the extensions Kn /K are geometric for all n (seeDefinition 2.14), a statement we conjecture to hold for p "= 2 but which we can only show whenp ≡ 3 mod 4 (Corollary 3.40). To illustrate Theorem 2.18, we give here a few values of δ(In ). We√√
can describe the n = 1 case completely: the roots of fx are { −x, − −x}, which we label {a1 , a2 }.

Clearly we have G1 = {e, (a1 a2 )}, whence δ(I1 ) = 1/2. In Example 2.19, page 29, we work out
the case n = 2, showing that δ(I2 ) = 3/8. Only with significantly more work (Theorem 3.2, (5.28),

and Corollary 5.11) can we show δ(I3 ) = 39/128. Moreover we note in Chapter 3 (see the discussion
on page 56) that for n > 7, δ(In ) may depend on the prime p, and cannot in general be easily
computed.
In Chapter 3 we undertake an analysis of the groups Gn . We do this through the study of
Hn = Gal (Kn /Kn−1 ). In (3.2), we show that Kn is obtained from Kn−1 by adjoining the squareroots of 2n−1 elements. Thus |Hn | ≤ 22

n−1, and we call Hn maximal if this inequality is an equality.

One principal result of the chapter is that for all p "= 2, Hn is maximal for n squarefree, andif p ≡ 3 mod 4 then Hn is maximal for all n (Theorem 3.2). The other result that is of central

importance in later chapters is Corollary 3.23 at the end of Section 3.2 (see below for explanation).
In Section 3.1, we prove some basic properties about fxn , and we introduce the polynomialspn ∈ Fp [x], defined by p1 = x and pn = p2n−1 + x for n ≥ 2, which play an important role. We
show that the discriminant of fxn is a product of powers of pi for i ≤ n (Proposition 3.9) and use

this to prove that Gn is not alternating, i.e. composed of even permutations (Corollary 3.10). InSection 3.2 we examine the center of Gn . We note that Gn is a 2-group, and we establish a series ofpropositions about 2-groups acting on certain sets. This culminates in Theorem 3.22, which states

5
that if Gn is not alternating then a certain permutation of the roots of fxn must lie in Gn . Indeed,as shown in Corollary 3.23, this permutation must lie in Hn , proving that Hn is nontrivial for all n.
In Section 3.3 we use abelian Kummer Theory to show that Hn is maximal if and only if pn isa square in Kn−1 (Theorem 3.27). This relies heavily on Corollary 3.23. In Section 3.4 we use thework of the previous sections to show that Hn is maximal if and only if a certain element Φn ∈ Fp [x]is a not a square in K = Fp (x) (Theorem 3.38). Specifically, Φn is the primitive part of pn :Φn ="(pd )µ(n/d) .
d|n

One can easily show that the degree of Φn is odd when n is squarefree (Corollary 3.30), establishing
the maximality of Hn for squarefree n. When n is not squarefree one cannot rule out the possibility
that Φn is a square in K. However, the facts that Φn is separable over Q for all n (see the proofof Theorem 3.2, page 54) and the degree of Φn grows like 2n suggest this is unlikely. We thus
conjecture that Hn is maximal for all n. In section 3.5 we adapt an argument of Stoll [35] to showthat if p ≡ 3 mod 4 then Hn is maximal for all n.

The results of Chapter 3 provide some insight into the structure of Hn and therefore Gn , butthere is no obvious way to use them to compute the limit as n → ∞ of the expression in (3). InChapters 4 and 5 we build a stochastic process where the main results of Chapter 3 have a naturalinterpretation. The tools of the theory of stochastic processes then allow us to prove this process is
eventually 0 with probability 1, and this is enough to establish Theorem 1.7.
Chapter 4 focuses on proving the process we seek exists and fleshing out definitions and standard
results from the theory of stochastic processes. In Section 4.1 we recall that a discrete-time stochastic
process (or simply process for short) is a sequence {Xn }n≥0 of random variables defined on a common

probability space. We consider only processes whose random variables take positive-integer values.
Such a process can be thought of as a game of chance, with Xn denoting a gambler’s score at turnn. We prove that there exists a process where the probability of the gambler having score t at turnn is determined by the structure of Gn . Specifically,
P(Xn = t) =1# {g ∈ Gn : g fixes t roots of fxn }
#Gn

(4)

We actually prove an even stronger property (see (4.3)). From (4) and the remark immediately
before (3) it follows that
δ(In ) = P(Xn > 0).

(5)

For more on why probability theory is a relatively natural tool in this context, see the introduction
to Chapter 4 on page 59.

6The main work of section 4.1 is to show that a process satisfying (4) exists. We call this process
the Galois process of the iterates of f , or GP(f ) for short. (Because there is no possibility ofambiguity, we drop the x from previous notation and simply write f = y 2 + x.) In Section 4.2 wegive some probabilistic background, including definitions of martingales and Markov chains and one
version of the basic martingale convergence theorem (Theorem 4.9). In Section 4.3 we present thebasic theory of branching processes, which not only is useful in Chapter 5 but also illustrates someof the definitions of Section 4.2.
In Chapter 5 all of the threads come together. In Section 5.1 we use Corollary 3.23, whichguarantees the existence of a certain type of element in each Hn , to establish that GP(f ) is amartingale. Thus GP(f ) is the first known example of a class of processes we call Galois martingales(see the remark on page 80), giving some justification for the first two words of this thesis’ title.
The martingale convergence theorem then shows that with probability 1 the sequence {Xn (ω)}n≥0
is eventually constant (where ω is any element in the underlying probability space).

In Section 5.2 we compute, under the assumption that Hn is maximal, the conditional distributionof Xn given Xn−1 = t for any value of t. Thus, using the metaphor of the gambler, when n is suchthat Hn is maximal we have explicit information about the probability that the gambler’s score goesfrom t to t% at turn n. Using Theorem 3.2, which says that Hn is maximal when n is squarefree, weshow that for any t ≥ 1 and m ≥ 1,P (Xn = t for all n ≥ m) = 0.
This is Theorem 5.8. It quickly follows that GP(f ) converges to 0 with probability 1. We deducefrom this in Section 5.3 that lim P(Xn > 0) = 0, and then from (5) and the remark followingn→∞(2) we conclude that δ(H(Fp )) = 0, which proves Theorem 1.7. Finally, we show that under theassumption that Hn is maximal for all n, GP(f ) is a particularly simple branching process. We also
give explicit values for δ(In ) under this assumption (see (5.28) and Corollary 5.11).