Recent Papers

with Su-Ion Ih,
"A finiteness property of postcritically finite unicritical polynomials,"
submitted. Available at ArXiv:2010.15941
Abstract: Let k be a number field with algebraic closure k and let S be a finite set of places of k containing all the archimedean ones. Fix d≥2 and α∈k such that the map zzd+α is not postcritically finite. Assuming a technical hypothesis on α, we prove that there are only finitely many parameters ck for which zzd+c is postcritically finite and for which c is S-integral relative to (α). That is, in the moduli space of unicritical polynomials of degree d, there are only finitely many PCF k-rational points that are ((α),S)-integral. We conjecture that the same statement is true without the technical hypothesis.

with Su-Ion Ih,
"Discreteness of postcritically finite maps in p-adic moduli space,"
submitted. Available at ArXiv:2005.04656
Abstract: Let p≥2 be a prime number and let Cp be the completion of an algebraic closure of the p-adic rational field Qp. Let fc(z) be a one-parameter family of rational functions of degree d≥2, where the coefficients are meromorphic functions defined at all parameters c in some open disk DCp. Assuming an appropriate stability condition, we prove that the parameters c for which fc is postcritically finite (PCF) are isolated from one another in the p-adic disk D, except in certain trivial cases. In particular, all PCF parameters of the family fc(z)=zd+c are p-adically isolated.

with Faseeh Ahmad, Jennifer Cain, Gregory Carroll, and Lily Fang,
"The arithmetic basilica: a quadratic PCF arboreal Galois group,"
Journal of Number Theory, accepted pending revisions. 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 x0K, 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) = z2−1 when the field K and root point x0 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 x0.

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 fL2(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 L2(Rd) 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 Rd.

with Jamie Juul,
"Odoni's conjecture for number fields,"
(Bulletin of the London Mathematical Society 51, #2 (2019), 237-250. Published online 2018; DOI: 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 fK[z] of degree d, and a point x0K, such that for every n≥0, the so-called arboreal Galois group Gal(K(fn(x0))/K) is an n-fold wreath product of the symmetric group Sd. In this paper, we prove Odoni’s conjecture when d is even and K is an arbitrary number field, and also when both d and [K:Q] are odd.

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:
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 fK(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 fQ(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 ĥf(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 fK(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 : P1P1 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.

Other Papers

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 fK(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 : PnPn.

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 Ap(u). The Björck sequences, which are defined on Z/pZ for p>2 prime, are unimodular and have zero autocorrelation on (Z/pZ)\{0}. These two properties give rise to the acronym, CAZAC, to refer to constant amplitude zero autocorrelation sequences. The bound proven is |Ap(u)| ≤ 2/p1/2 + 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 1K 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=Qp, the field of p-adic rational numbers (as a group under addition), which has the compact open subgroup H=Zp, 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 f1,...,fg be rational functions in C(z), let F=(f1,...,fg) denote their coordinatewise action on (P1)g, let V be a proper subvariety of (P1)g, and let P=(x1,...,xg) ∈(P1)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 Fn(P)∈V(C) must be very sparse. In particular, for any k and any sufficiently large N, the number of n<N such that Fn(P)∈V(C) is less than logkN, where logk 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 Pg has only finite intersection with any curve contained in Pg. 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 slog(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 Cp to any algebraically closed complete non-archimedean field CK 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 CK[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 fK(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=Qp, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H=Zp, 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 Qp of p-adic rational numbers, with subgroup Zp), and of various wavelets on those groups.

"Examples of wandering domains in p-adic polynomial dynamics"
(Comptes Rendus Mathématique. Académie des Sciences. Paris , 335 (2002), 615--620.)
Abstract: For any prime p>0, we construct p-adic polynomial functions in Cp[z] whose Fatou sets have wandering domains.

"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 fK(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)=z2+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"
(Compositio Mathematica 122:3 (2000), 281--298.)
Abstract: In this paper we study dynamics on the Fatou set of a rational function f(z) defined over a finite extension Qp, the field of p-adic rationals. Using a notion of ``components'' of the Fatou set defined in ``Hyperbolic Maps in p-adic Dynamics'' (below), we state and prove an analogue of Sullivan's No Wandering Domains Theorem for p-adic rational functions which have no wild recurrent Julia critical points.

"Hyperbolic maps in p-adic dynamics"
(Ergodic Theory and Dynamical Systems 21 (2001), 1--11.)
Abstract: In this paper we study the dynamics of a rational function f(z) defined over a finite extension Qp, the field of p-adic rationals. After proving some basic results, we define a notion of ``components'' of the Fatou set, analogous to the topological components of a complex Fatou set. We define hyperbolic p-adic maps and, in our main theorem, characterize hyperbolicity by the location of the critical set. We use this theorem and our notion of components to state and prove an analogue of Sullivan's No Wandering Domains Theorem for hyperbolic maps.

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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