with Faseeh Ahmad, Jennifer Cain, Gregory Carroll, and Lily Fang,

"The arithmetic basilica: a quadratic PCF arboreal Galois group,"

(submitted).
Available at
ArXiv:1909.00039)

**Abstract:**
The arboreal Galois group of a polynomial *f* over a field *K*
encodes the action of Galois on the iterated preimages of a root point
*x*_{0}∈*K*, analogous to the action of Galois
on the *l*-power torsion of an abelian variety. We compute
the arboreal Galois group of the postcritically finite polynomial
*f*(*z*) = *z*^{2}−1
when the field *K* and root point *x*_{0}
satisfy a simple condition. We call the resulting group the
*arithmetic basilica group* because of its relation to the
basilica group associated with the complex dynamics of *f*.
For *K* = **Q**, our condition
holds for infinitely many choices of *x*_{0}.

with Patrick Ingram, Rafe Jones, Michelle Manes,
Joe Silverman, and Tom Tucker,

"Current trends and open problems in arithmetic dynamics,"

(*Bulletin of the American Mathematical Society (N.S.)*
**56**, #4 (2019), 611-685.)
Also available at
ArXiv:1806.04980)

**Abstract:**
Arithmetic dynamics is the study of number theoretic properties
of dynamical systems. A relatively new field, it draws inspiration
partly from dynamical analogues of theorems and conjectures
in classical arithmetic geometry, and partly from *p*-adic
analogues of theorems and conjectures in classical complex dynamics.
In this article we survey some of the motivating problems and
some of the recent progress in the field of arithmetic dynamics.

with John J. Benedetto,

"Frames of translates of number-theoretic groups,"
,

*Journal of Geometric Analysis*, 2019.
Online First: DOI 10.1007/s12220-019-00234-y .
Also available at
ArXiv:1804.07783

**Abstract:**
Frames of translates of *f*∈*L*^{2}(*G*)
are characterized in terms of the zero-set of the so-called
spectral symbol of *f* in the setting of a
locally compact abelian group *G* having a compact open subgroup
*H*. We refer to such a *G* as a number theoretic group.
This characterization was first proved in 1992 by Shidong Li
and one of the authors for
*L*^{2}(**R**^{d})
with the same *formal* statement of the characterization.
For number theoretic groups, and these include local fields,
the strategy of proof is necessarily entirely different;
and it requires a new notion of translation that reduces
to the usual definition in **R**^{d}.

with Jamie Juul,

"Odoni's conjecture for number fields,"
,

(*Bulletin of the London Mathematical Society* **51**,
#2 (2019), 237-250.
Published online 2018; DOI:
https://doi.org/10.1112/blms.12225
Also available at
ArXiv:1803.01987)

**Abstract:**
Let *K* be a number field, and let *d*≥2.
A conjecture of Odoni (stated more generally for characteristic zero
Hilbertian fields *K*) posits that there is a monic polynomial
*f*∈*K*[*z*] of degree *d*,
and a point *x*_{0}∈*K*,
such that for every *n*≥0, the so-called arboreal Galois group
Gal(*K*(*f*^{−n}(*x*_{0}))/*K*)
is an *n*-fold wreath product of the symmetric group
*S _{d}*. In this paper, we prove Odoni’s conjecture
when

with Xander Faber, Benjamin Hutz, Jamie Juul, and Yu Yasufuku,

"A large arboreal Galois representation for a cubic postcritically
finite polynomial,"

(*Research in Number Theory* **3**, #29 (2017).
Open access:
https://doi.org/10.1007/s40993-017-0092-8)

**Abstract:**
We give a complete description of the arboreal Galois representation
of a certain postcritically finite cubic polynomial over a large class
of number fields and for a large class of basepoints. The associated
Galois action on an infinite ternary rooted tree has Hausdorff
dimension bounded strictly between that of the infinite wreath product
of cyclic groups and that of the infinite wreath product of symmetric
groups. We deduce a zero-density result for prime divisors in an orbit
under this polynomial. We also obtain a zero-density result for the
set of places of convergence of Newton's method for a certain cubic
polynomial, thus resolving the first nontrivial case of a conjecture
of Faber and Voloch.

with Dvij Bajpai, Ruqian Chen, Edward Kim, Owen Marschall,
Darius Onul, and Yang Xiao,

"Non-archimedean connected Julia sets with branching"

(*Ergodic Theory Dynam. Systems*
**37** (2017), 59-78.)

**Abstract:**
We construct the first examples of rational functions
defined over a nonarchimedean field with a certain
dynamical property: the Julia set in the Berkovich projective line
is connected but not contained in a line segment.
We also show how to compute the measure-theoretic
and topological entropy of such maps. In particular,
we give an example for which
the measure-theoretic entropy is strictly
smaller than the topological entropy,
thus answering a question of Favre and Rivera-Letelier.

"Attaining Potentially Good Reduction
in Arithmetic Dynamics"

(*International Mathematics Research Notices*
**2015**, #22, 11828-11846.)

**Abstract:**
Let *K* be a non-archimedean field, and
let *f*∈*K*(*z*)
be a rational function of degree *d*≥2.
If *f* has potentially good reduction, we give an upper
bound, depending only on *d*, for the minimal degree of
an extension *L*/*K* such that *f*
is conjugate over *L* to a map of good reduction.
In particular, if *d*=2 or *d* is
greater than the residue characteristic of *K*,
the bound is *d*+1.
If *K* is discretely valued, we give examples to
show that our bound is sharp.

with Ruqian Chen, Trevor Hyde, Yordanka Kovacheva, and Colin White,

"Small Dynamical Heights for Quadratic Polynomials and Rational
Functions"

(*Experimental Mathematics* **23** (2014), 433-447.)

**Abstract:**
Let *f*∈**Q**(*z*)
be a polynomial or rational function of degree 2.
A special case of Morton and Silverman's
Dynamical Uniform Boundedness Conjecture states that
the number of rational preperiodic points of *f*
is bounded above by an absolute constant.
A related conjecture of Silverman states that the canonical height
*ĥ*_{f}(*x*)
of a non-preperiodic rational point *x* is
bounded below by a uniform multiple of the height of *f* itself.
We provide support for these conjectures by computing the set
of preperiodic and small height rational points
for a set of degree 2 maps far beyond the range of previous searches.

"A Criterion for Potentially Good Reduction
in Non-archimedean Dynamics"

(*Acta Arithmetica* **165** (2014), 251-256.)

**Abstract:**
Let *K* be a non-archimedean field,
and let *f*∈*K*(*z*)
be a polynomial or rational function of degree at least 2.
We present a necessary and sufficient condition,
involving only the fixed points of *f* and their preimages,
that determines whether or not the dynamical system
*f* : **P**^{1} → **P**^{1}
has potentially good reduction.

with
Patrick Ingram,
Rafe Jones,
and
Alon Levy,

"Attracting cycles in *p*-adic dynamics
and height bounds for postcritically finite maps"

(*Duke Mathematical Journal* **163** (2014), 2325-2356.)

**Abstract:**
A rational function *f*(*z*)∈**C**(*z*)
of degree *d*≥2 is postcritically finite (PCF)
if all of its critical points have finite forward orbit
under iteration of *f*. We show that the collection
of PCF rational functions is a set of bounded height in
moduli space, once the well-understood family known as flexible
Lattès maps is excluded.
As a consequence, there are only finitely many conjugacy classes
of non-Lattès PCF rational maps of a given degree defined over any
given number field.
The key ingredient of the proof is a non-archimedean version
of Fatou's classical result that every attracting cycle
of a rational function over **C** attracts a critical point.

with
Dragos Ghioca,
Pär Kurlberg,
Ben Hutz,
Thomas Scanlon,
and
Tom Tucker,

"Periods of rational maps modulo primes"

(*Mathematische Annalen* **355** (2013), 637-660.)

**Abstract:**
Let *K* be a number field, let
*f*∈*K*(*t*) be a rational map
of degree at least 2, and let *a* , *b* be points in *K* .
We show that if *a* is not in the forward orbit of *b*,
then there is a
positive proportion of primes **p** of *K* such that
*a* (mod **p**) is not in the forward orbit of
*b* (mod **p**).
Moreover, we show that a similar result holds for several maps and
several points.
We also present heuristic and numerical evidence that a higher
dimensional analog of this result is unlikely to be true if we replace
*a* by a hypersurface, such as the ramification locus of a
morphism *f* : **P**^{n} →
**P**^{n}.

with
John J. Benedetto
and Joseph T. Woodworth,

"Optimal ambiguity functions and Weil's exponential sum bound"

(*Journal of Fourier Analysis and Applications*
**18** (2012), 471-487.)

**Abstract:**
Complex-valued periodic sequences,
*u*, constructed by Göran Björck,
are analyzed with regard to the behavior of their discrete periodic
narrow-band ambiguity functions *A*_{p}(*u*).
The Björck sequences, which are defined on
**Z**/*p***Z** for *p*>2 prime,
are unimodular and have zero autocorrelation on
(**Z**/*p***Z**)\{0}.
These two properties give rise to the
acronym, CAZAC, to refer to constant amplitude zero autocorrelation
sequences. The bound proven is
|*A*_{p}(*u*)| ≤ 2/*p*^{1/2}
+ 4/*p* outside of (0,0),
and this is of optimal magnitude given the constraint that *u*
is a CAZAC sequence. The proof requires the full power of Weil's
exponential sum bound, which, in turn, is a consequence of his proof of
the Riemann hypothesis for finite fields.
Such bounds are not only of
mathematical interest, but they have direct applications as sequences in
communications and radar, as well as when the sequences are used as
coefficients of phase-coded waveforms.

with
John J. Benedetto,

"The construction of wavelet sets"

(in Jonathan Cohen, Ahmed I. Zayed, eds.,
*Wavelets and Multiscale Analysis: Theory and Applications*
Springer, New York, 2011; pages 17-56.)

**Abstract:**
Sets *K* in *d*-dimensional Euclidean space are constructed with
the property that the inverse Fourier transform of the characteristic
function **1**_{K} of the set *K* is a single dyadic
orthonormal wavelet. The iterative construction is characterized by
its generality, its computational implementation, and its
simplicity. The construction is transported to the case of locally
compact abelian groups *G* with compact open subgroups *H*.
The best known example of such a group is
*G*=**Q**_{p}, the field of
*p*-adic rational numbers (as a group under addition), which has the
compact open subgroup *H*=**Z**_{p},
the ring of *p*-adic
integers. Fascinating intricacies arise. Classical wavelet theories,
which require a non-trivial discrete subgroup for translations, do not
apply to *G*, which may not have such a subgroup.
However, our wavelet theory is formulated on *G*
with new group theoretic operators, which can be thought of
as analogues of Euclidean translations. As such, our theory
for *G* is structurally cohesive and of significant generality.
For perspective, the Haar and
Shannon wavelets are naturally antipodal in the Euclidean setting,
whereas their analogues for *G* are equivalent.

with
Dragos Ghioca,
Pär Kurlberg, and
Tom Tucker,

"A gap principle for dynamics"

(*Compositio Mathematica* **146** (2010), 1056-1072.)

**Abstract:**
Let *f*_{1},...,*f*_{g} be
rational functions in **C**(*z*), let
*F*=(*f*_{1},...,*f*_{g})
denote their coordinatewise action on
(**P**^{1})^{g},
let *V* be a proper subvariety of
(**P**^{1})^{g}, and
let *P*=(*x*_{1},...,*x*_{g})
∈(**P**^{1})^{g}(**C**) be
a nonpreperiodic point for *F*.
We show that if *V* does not contain any periodic
subvarieties of positive dimension, then the set of *n* such that
*F*^{n}(*P*)∈*V*(**C**)
must be very sparse. In particular, for any
*k* and any sufficiently large *N*,
the number of *n*<*N* such that
*F*^{n}(*P*)∈*V*(**C**)
is less than log^{k}*N*, where
log^{k}
denotes the *k*-th iterate of the log function. This can be
interpreted as an analog of the gap principle of Davenport-Roth and
Mumford.

with
Ben Dickman, Sasha Joseph, Ben Krause, Dan Rubin, and Xinwen Zhou,

"Computing points of small height for cubic polynomials"

(*Involve* **2** (2009), 37-64.)

**Abstract:**
Let *f* be a polynomial of degree *d* at least two
in **Q**[*z*]. The associated canonical height is a
certain real-valued function on **Q** that returns zero
precisely at preperiodic rational points of *f*. Morton
and Silverman conjectured in 1994 that the number of such points
is bounded above by a constant depending only on *d*.
A related conjecture claims that at non-preperiodic rational
points, the canonical height is bounded below by a positive
constant (depending only on *d*) times some kind of height
of *f* itself. In this paper, we provide support for these
conjectures in the case *d*=3 by computing the set of
small height points for several billion cubic polynomials.

"Review: *The Arithmetic of Dynamical Systems*,
by Joseph H. Silverman"

(*Bulletin of the American Mathematical Society*
**46** (2009), 157-164.)

with
Dragos Ghioca,
Pär Kurlberg, and
Tom Tucker,

with an appendix by
Umberto Zannier,

"A Case of the Dynamical Mordell-Lang Conjecture"

(*Mathematische Annalen* **352** (2012), 1-26.)

Link to published version here
or
here

**Abstract:**
We prove a special case of a dynamical analogue of the classical
Mordell-Lang conjecture. In particular, let *f* be a rational
function with no superattracting periodic points other than
exceptional points. If the coefficients of *f* are algebraic, we
show that the orbit of a point outside the union of proper
preperiodic subvarieties of **P**^{g}
has only finite intersection with any curve contained in
**P**^{g}. Our proof uses results
from *p*-adic dynamics together with an integrality argument.

with
Liang-Chung Hsia,

"A quotient of elliptic curves - weak Néron models for
Lattès maps"

(Proceedings of the
2007 Waseda Number Theory Symposium.)

**Abstract:**
A Lattès map is a morphism of the projective line (i.e., a rational
function in one variable) induced as a quotient of an endomorphism of an
elliptic curve. We present an algorithm for constructing a weak
Néron model for such a map from a quotient of a
Néron model of the elliptic curve, at least for non-archimedean
fields of residue characteristic not equal to 2. We defer the proofs
to a future paper.

with
Jean-Yves Briend and
Hervé Perdry,

"Dynamique des polynômes quadratiques sur les corps locaux"

(*Journal de Théorie des Nombres de Bordeaux*
**19** (2007), 325-336.)

**Abstract:**
We show that the dynamics of a quadratic polynomial over a local field
can be completely decided in a finite amount of time, with the following
two possibilities: either the Julia set is empty, or the polynomial is
topologically conjugate on its Julia set to the one-sided shift on two
symbols.

"Heights and preperiodic points of polynomials over function fields"

(*International Mathematics Research Notices*,
**2005**, #62, 3855-3866.)

**Abstract:**
Let *K* be a function field in one variable over an arbitrary
field *F*. Given a rational function
*f*(*z*)∈*K*(*z*)
of degree at least two, the associated canonical
height on the projective line was defined by Call and Silverman.
The preperiodic points of *f* all have canonical height zero;
conversely, if *F* is a finite field, then every point of
canonical height zero is preperiodic. However, if *F* is an
infinite field, then there may be non-preperiodic points of canonical
height zero. In this paper, we show that for polynomial
*f*, such points exist only if *f* is isotrivial.
In fact, such *K*-rational points exist only if *f* is
defined over the constant field of *K*
after a *K*-rational change of coordinates.

"Preperiodic points of polynomials over global fields"

(*Journal für die Reine und Angewandte Mathematik*
**608** (2007), 123-153.)

**Abstract:**
Given a global field *K* and a polynomial *f* defined
over *K* of degree at least two, Morton and Silverman
conjectured in 1994 that the number of *K*-rational
preperiodic points of *f* is bounded
in terms of only the degree of *K* and the degree of *f*.
In 1997, for quadratic polynomials over *K*=**Q**,
Call and Goldstine proved a bound which was exponential in *s*,
the number of primes of bad reduction of *f*.
By careful analysis of the filled Julia sets at each prime,
we present an improved bound on the order of *s*log(*s*).
Our bound applies to polynomials of any degree (at least two)
over any global field *K*.

"An Ahlfors Islands Theorem for Non-archimedean Meromorphic Functions"

(*Transactions of the American Mathematical Society*
**360** (2008), 4099-4124.)

**Abstract:**
We present a *p*-adic and non-archimdean version of the
Five Islands Theorem for meromorphic functions from
Ahlfors' theory of covering surfaces. In the non-archimedean
setting, the theorem requires only four islands, with explicit
constants. We present examples to show that the constants are
sharp and that other hypotheses of the theorem cannot be removed.
This paper extends an earlier theorem of the author for holomorphic
functions.

"Wandering Domains in Non-Archimedean Polynomial Dynamics"

(*Bulletin of the London Mathematical Society*,
**38** (2006), 937-950.)

**Abstract:**
We extend a recent
result on the existence of wandering domains
of polynomial functions defined over the *p*-adic
field **C**_{p} to
any algebraically closed complete non-archimedean field
**C**_{K}
with residue characteristic *p*> 0.
In fact,
we prove polynomials with wandering domains form
a dense subset of a certain one-dimensional family
of degree *p*+1 polynomials in
**C**_{K}[z].

"Wandering Domains and Nontrivial Reduction in Non-Archimedean Dynamics"

(*Illinois Journal of Mathematics* **49** (2005), 167-193.)

**Abstract:**
Let *K* be a non-archimedean field with residue field *k*,
and suppose that *k* is not an algebraic extension of a
finite field. We prove two results concerning wandering
domains of rational functions *f*∈*K*(*z*) and
Rivera-Letelier's notion of nontrivial reduction.
First, if *f* has nontrivial reduction, then assuming some
simple hypotheses, we show that the Fatou set of
*f* has wandering components by any of the usual
definitions of ``components of the Fatou set''. Second,
we show that if *k* has characteristic zero and *K*
is discretely valued, then the existence of a wandering
domain implies that some iterate has nontrivial reduction
in some coordinate.

with
John J. Benedetto,

"A wavelet theory for local fields and related groups"

(*The Journal of Geometric Analysis* **14** (2004), 423-456.)

**Abstract:**
Let *G* be a locally compact abelian group with compact open
subgroup *H*.
The best known example of such a group is
*G*=**Q**_{p}, the
field of *p*-adic rational numbers
(as a group under addition), which has compact open subgroup
*H*=**Z**_{p}, the ring of *p*-adic integers.
Classical wavelet theories, which require a non-trivial discrete subgroup
for translations, do not apply to *G*,
which may not have such a subgroup.
A wavelet theory is developed on *G*
using coset representatives of the discrete quotient
of the dual of *G* by the annihilator of *H*
to circumvent this limitation.
Wavelet bases are constructed by means of an iterative method
giving rise to so-called wavelet sets in the dual group *G*.
Although the Haar and Shannon wavelets are naturally antipodal
in the Euclidean setting,
it is observed that their analogues for *G* are equivalent.

"Examples of Wavelets for Local Fields"

(*Contemporary Mathematics* **345**, AMS, Providence, 2004,
pages 27-47.)

**Abstract:**
Let *G* be a locally compact abelian group with a compact open
subgroup *H*. Given an expansive
automorphism *A* of
*G*, J. Benedetto and the author have
proposed a theory of wavelets on *G*, including the construction
of wavelet sets. In this expository paper, we consider
some specific examples of the wavelet theory on such groups.
In particular,
we show that Shannon wavelets on *G* are the same as Haar wavelets
on *G*. We give several examples of specific groups (such
as the additive group **Q**_{p}
of *p*-adic rational numbers,
with subgroup **Z**_{p}),
and of various wavelets on those groups.

**
"Examples of wandering domains in p-adic polynomial
dynamics"
**

(

**
"Non-archimedean holomorphic maps and the Ahlfors Islands Theorem"
**

(*American Journal of Mathematics*, **125** (2003), 581--622.)

**Abstract:**
We present a *p*-adic and non-archimedean version of some
classical complex holomorphic function theory. Our main result
is an analogue of the Five Islands Theorem from Ahlfors' theory
of covering surfaces. For non-archimedean holomorphic maps,
our theorem requires only two islands, with explicit and nearly
sharp constants, as opposed to the three islands without explicit
constants in the complex holomorphic theory. We also present
non-archimedean analogues of other results from the complex
theory, including theorems of Koebe, Bloch, and Landau, with
sharp constants.

**
"Components and periodic points in non-archimedean dynamics"
**

(*Proceedings of the London Mathematical Society* (3)
**84** (2002), 231--256.)

**Abstract:**
We expand the notion of non-archimedean connected
components introduced in *Hyperbolic maps in p-adic dynamics*
(see below). We define two types of components
and discuss their uses and applications in the study of dynamics
of a rational function *f*∈*K*(*z*)
defined over a non-archimedean field *K*.
Using this theory, we derive several results on the geometry
of such components and the existence of periodic points within them.
Furthermore, we demonstrate that for appropriate fields of
definition, the conjectures stated in
*p-adic dynamics and Sullivan's No Wandering Domains
Theorem* (see below),
including the No Wandering Domains conjecture,
are equivalent regardless of which definition of ``component''
is used. We also give a number of examples of *p*-adic
maps with interesting or pathological dynamics.

**
"An elementary product identity in polynomial dynamics"
**

(*The American Mathematical Monthly* **108** (2001), 860--864.)

**Abstract:** Given a quadratic polynomial of the form
*f*(*z*)=*z*^{2}+c and a periodic
cycle of *f* of period at least 2, we demonstrate
that the certain sums of points in the cycle have product
1. We generalize our identity to any monic polynomial
with any two distinct periodic points. The proof turns
out to be simple and elementary. We also use our
identity to produce algebraic units over an integral
domain.

(PDF reprint version available here.)

**
"Reduction, dynamics, and Julia sets of rational functions"
**

(*The Journal of Number Theory* **86** (2001), 175--195.)

**Abstract:**We consider a rational function
*f*(*z*)∈*K*(*z*) in one
variable defined over an algebraically closed field *K*
which is complete with respect to a valuation *v*.
We study how the reduction (modulo *v*) of such
functions behaves under composition, and in particular
under iteration. We also investigate the relationship
between bad reduction and the Julia set of *f*.
In particular, we prove that under certain conditions,
bad reduction is equivalent to having a
nonempty Julia set. We also give several examples
of maps not satisfying those conditions and having
both bad reduction and empty Julia set.

**
" p-adic dynamics and Sullivan's No Wandering Domains theorem"
**

(

**
"Hyperbolic maps in p-adic dynamics"
**

(

with William Goldman,

**
``The topology of the relative character varieties
of a quadruply-punctured sphere''
**

*Experiment. Math.* **8**:1 (1999),
85--103.

"Fatou Components in *p*-adic dynamics"

(Brown University, 1998.)

**Abstract:**
We study the dynamics of a rational function *f* defined
over the *p*-adic numbers and acting on the *p*-adic projective
line. Using the theory of complex dynamics as a model, we
define the Fatou and Julia sets of such a function and study
their properties. We define two notions of "connected
components" of the Fatou set appropriate to the non-Archimedean (and
therefore totally disconnected) setting. Using these notions,
we state and prove a partial analogue of Dennis Sullivan's
No Wandering Domains Theorem and related results.

Note 1: There are a few mathematical errata I know of in the original thesis. Download the short text file errata.txt here for a list and description.

Note 2: Most of the results of my thesis appeared in
the three papers
"Hyperbolic maps in *p*-adic dynamics",
"*p*-adic dynamics and Sullivan's No Wandering Domains theorem",
and
"Reduction, dynamics, and Julia sets of rational functions"
listed above, though the third paper also included a number
of other results.
The analysis of quadratic Julia sets in Section 3.3 and Appendix A
was never published, but generalizations of those results (to a
larger class of base fields) with far cleaner proofs have appeared
in the paper
"Dynamique des polynômes quadratiques sur les corps locaux"
above.
A few smaller thesis results, like Theorem 3.1.3 (bounding
the number of times the preimage of a disk includes non-disks),
the construction
of an entire function with a wandering domain in Section 5.5,
and
the cubic polynomial examples computed in Section 7.2,
have never been published.

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