Wandering Domains in Non-Archimedean Polynomial Dynamics

Abstract

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. 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].

Invariant manifolds for analytic dynamical systems over ultrametric fields

“…3 Complementary to our studies, much of the literature on ultrametric dynamical systems can rather be regarded as an offspring of complex dynamics, and has concentrated on the 1-dimensional case (see, e.g., [6] and [9]). Some specific new phenomena arose there, like the existence of wandering domains [7]. It also turned out to be necessary in some situations to extend the action of polynomials or rational functions from the ordinary projective line to the Berkovich projective line, because the latter supports relevant measures while the projective line does not, in contrast to the classical complex case [14] (further ultrametric phenomena can be found in [4] and [32].…”

“…Condition(7) is automatically satisfied if we take A := f ′ (x) and choose r > 0 small enough, since the analytic map f is "strictly differentiable" at x and thus lim s→0 Lip( f | B E s (x) ) = 0 (see 4.2.3 and 3.2.4 in[10]). …”

“…For the single-variable case, consult[34] and the references therein 6. These are also needed to adapt the above Lie-theoretic results to non-analytic C k -Lie groups or non-analytic C k -automorphisms, as constructed in[16] 7. With some precautions if k = 1 and the modelling space is infinite-dimensional.…”

Primitive divisors in arithmetic dynamics

“…The answer is probably not. Rob Benedetto (private communication) has sketched an argument using ideas from [4,6] which suggests that there are dynamical examples over Q p with γ periodic such that ord p A n grows extremely rapidly, for example faster than O(d n ), or even faster than O(2 d n ). This is in marked contrast to the situation over a number field, where the elementary height estimate (1) shows that ord p ϕ n (α) cannot grow faster than O(d n ).…”

On hyperbolic fixed points in ultrametric dynamics

“…Recall that by (9) we also have δ f ≥ |λ|ρ. It follows that the radius of injectivity δ f = |λ|ρ if ρ = 1/|a 2 |.…”