Robert L. Benedetto

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 = ℚp, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = ℤp, the ring of padic 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 Ĝ/H⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.

Hyperbolic maps in p -adic dynamics

In this paper we study the dynamics of a rational function \phi\in K(z) defined over some finite extension K of \mathbb{Q}_ p . 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 Sullivans No Wandering Domains Theorem for hyperbolic maps.

p-Adic Dynamics and Sullivan's No Wandering Domains Theorem

AbstractIn this paper we study dynamics on the Fatou set of a rational function ϕ ∈

Examples of wandering domains in p-adic polynomial dynamics

\overline {\mathbb {Q}} _P (z)

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

. Using a notion of ‘components’ of the Fatou set defined by Benedetto, we state and prove an analogue of Sullivans No Wandering Domains Theorem for p-adic rational functions which have no wild recurrent Julia critical points.

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

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.

A gap principle for dynamics

Abstract For any prime p >0, we contruct p-adic polynomial functions in C p [z] whose Fatou sets have wandering domains. To cite this article: R.L. Benedetto, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 615–620.

Wandering Domains in Non-Archimedean Polynomial Dynamics

Abstract Given a global field K and a polynomial defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational pre-periodic points of is bounded in terms of only the degree of K and the degree of . 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 . 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.