"Review: The Arithmetic of Dynamical Systems, by Joseph H. Silverman,"
(Bulletin of the American Mathematical Society, to appear.)

with Dragos Ghioca, Pär Kurlberg, and Tom Tucker,
"The Dynamical Mordell-Lang Conjecture"
submitted.
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,
(PS) "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,
(PDF) "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) in 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 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 in 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 Benedetto,
(PS) "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"
(Comptes Rendus Mathématique. Académie des Sciences. Paris , 335 (2002), 615--620.)
Abstract: For any prime p>0, we construct p-adic polynomial functions in C_p[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 f in 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)=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) in 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 Q_p, 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 Q_p, 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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