arXiv subjects

Robert L. Benedetto

Publications and source records attributed to Robert L. Benedetto.

32 records · Page 2Linked to original sources

arXiv2013

Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps

A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes 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 the complex numbers attracts a critical point.

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arXiv2013

A Criterion for Potentially Good Reduction in Non-archimedean Dynamics

Let K be a non-archimedean field, and let f in 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 on P^1 has potentially good reduction.

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arXiv2011

Periods of rational maps modulo primes

Let $K$ be a number field, let $ϕ\in K(t)$ be a rational map of degree at least 2, and let $α, β\in K$. We show that if $α$ is not in the forward orbit of $β$, then there is a positive proportion of primes ${\mathfrak p}$ of $K$ such that $α\mod {\mathfrak p}$ is not in the forward orbit of $β\mod {\mathfrak p}$. 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 $α$ by a hypersurface, such as the ramification locus of a morphism $ϕ: {\mathbb P}^{n} \to {\mathbb P}^{n}$.

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arXiv2011

Optimal ambiguity functions and Weil's exponential sum bound

Complex-valued periodic sequences, u, constructed by Goran Bjorck, are analyzed with regard to the behavior of their discrete periodic narrow-band ambiguity functions A_p(u). The Bjorck 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 |A_p(u)| \leq 2/\sqrt{p} + 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.

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arXiv2009

The Dynamical Mordell-Lang Conjecture

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $ϕ$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $ϕ$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(\bP^1)^g$ has only finite intersection with any curve contained in $(\bP^1)^g$. We also show that our result holds for indecomposable polynomials $ϕ$ with coefficients in $\bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $\bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(ϕ,ϕ)$ on $\bA^2$.

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arXiv2008

Computing points of small height for cubic polynomials

Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height \hat{h}_f 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, \hat{h}_f 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.

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arXiv2008

A gap principle for dynamics

Let $f_1,...,f_g\in {\mathbb C}(z)$ be rational functions, let $Φ=(f_1,...,f_g)$ denote their coordinatewise action on $({\mathbb P}^1)^g$, let $V\subset ({\mathbb P}^1)^g$ be a proper subvariety, and let $P=(x_1,...,x_g)\in ({\mathbb P}^1)^g({\mathbb C})$ be a nonpreperiodic point for $Φ$. We show that if $V$ does not contain any periodic subvarieties of positive dimension, then the set of $n$ such that $Φ^n(P) \in V({\mathbb C})$ must be very sparse. In particular, for any $k$ and any sufficiently large $N$, the number of $n \leq N$ such that $Φ^n(P) \in V({\mathbb C})$ 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.

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arXiv2006

Wandering domains in non-archimedean polynomial dynamics

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. We also prove that polynomials with wandering domains form a dense subset of a certain one-dimensional family of degree p+1 polynomials in C_K[z].

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arXiv2005

Heights and preperiodic points of polynomials over function fields

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.

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arXiv2005

A wavelet theory for local fields and related groups

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 a quotient of the dual group of G. Wavelet bases are constructed by means of an iterative method giving rise to so-called wavelet sets in the dual group.

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arXiv2005

Preperiodic points of polynomials over global fields

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.

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arXiv2004

Wandering domains and nontrivial reduction in non-archimedean dynamics

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 such components. Second, we show that if k has characteristic zero and K is discretely valued, then the converse holds; that is, the existence of a wandering domain implies that some iterate has nontrivial reduction in some coordinate.

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arXiv2004

An Ahlfors Islands Theorem for non-archimedean meromorphic functions

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.

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arXiv2003

Examples of wavelets for local fields

Using the wavelet theory introduced by the author and J. Benedetto, we present examples of wavelets on p-adic fields and other locally compact abelian groups with compact open subgroups. We observe that in this setting, the Haar and Shannon wavelets (which are at opposite extremes over the real numbers) coincide and are localized both in time and in frequency. We also study the behavior of the translation operators required in the theory.

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