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 Cp to any algebraically closed complete non-archimedean field CK 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 CK [z]. Given a rational function φ ∈ K(z) with coefficients in a field K, one may consider the dynamical system given by the action of the iterates φ on P(K) = K ∪ {∞}, for n ≥ 0. Here, φ denotes the n-fold composition φ ◦ · · · ◦ φ, so that φ is the identity function, φ = φ, φ = φ ◦ φ, and so on. The case of complex dynamics, when K = C, has been studied intensively for several decades; see [1, 9, 15] for expositions. In particular, one may study the action of {φ} on the Fatou set F = Fφ, to be defined below. It is well known that the connected components of F are mapped onto one another by φ and that, according to Sullivan’s deep No Wandering Domains Theorem [20], every complex Fatou component is preperiodic under application of φ. It is also possible to define Fatou sets for other metric fields. The study of the resulting dynamics has seen growing interest in the past decade or two; see, for example, [2, 8, 11, 12, 14, 17]. In this paper we will study wandering domains over certain non-archimedean fields, and we fix the following notation. CK a complete and algebraically closed non-archimedean field | · | the absolute value on CK k̂ the residue field of CK p the residue characteristic char k̂ P(CK) the projective line CK ∪ {∞} We will assume throughout that p > 0; that is, that CK has positive residue characteristic. Note that 0 ≤ |p| < 1, when p is viewed as an element of CK under the unique nontrivial homomorphism of Z into CK . It is possible that charCK = 0 with char k̂ = p > 0; but if charCK > 0, then char k̂ = charCK . Recall that “non-archimedean” means CK satisfies the ultrametric triangle inequality |x+ y| ≤ max{|x|, |y|} for all x, y ∈ CK . If |x| 6 = |y|, it is immediate that |x + y| = max{|x|, |y|}. Note that |n| ≤ 1 for all n ∈ Z. Recall also that the residue field k̂ is defined to be OK/MK , where OK is the ring {x ∈ CK : |x| ≤ 1} of integers in CK , and MK is the maximal ideal of {x ∈ K : |x| < 1} of OK . It is easy to check that k̂ is algebraically closed because CK is. Date: November 8, 2003; revised February 10, 2006. 2000 Mathematics Subject Classification. Primary: 12J25; Secondary: 37F99. The author gratefully acknowledges the support of NSF grant DMS-0071541.