Consider an arbitrary set S and an arbitrary function f : R → S.
Abstract. We construct the first examples of rational functions defined over a nonarchimedean field with a certain dynamical property: the Julia set in the Berkovich projective line is connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we give an example for which the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.Let C v be an algebraically closed field equipped with a non-archimedean absolute value We will assume that C v is complete with respect toThe degree of a rational function φ ∈ C v (z) is deg φ := max{deg f, deg g}, where φ = f /g and f, g ∈ C v [z] are relatively prime polynomials. For n ≥ 0, we denote the n-th iterate of φ under composition by φ n ; that is, φ 0 (z) = z, and φ n+1 = φ • φ n .The rational function φ acts naturally on the Berkovich projective line P 1 Ber . We will discuss P 1Ber in more detail in Section 1; for the moment, we note only that P 1 Ber is a certain path-connected compactification of P 1 (C v ), and that many of the extra points correspond to closed disks in C v . We also note that the set H Ber := P 1 Ber P 1 (C v ) has a natural metric d H , although the metric topology on H Ber is stronger than the topology inherited from P 1 Ber . We call H Ber hyperbolic space, and d H the hyperbolic metric.The dynamical action of φ partitions P 1 Ber into two invariant subsets: the (Berkovich) Julia set J φ , which is the closed subset on which φ n acts chaotically, and the (Berkovich) Fatou set F φ , which is the (open) complement of J φ . There are numerous examples of Berkovich Julia sets in the literature; see, for example, [2,8,9,12,18,20,21,22]. However, in all these examples, the Julia set is either a single point, a line segment, or a disconnected set (in which case it necessarily has infinitely many connected components). In this paper, we give the first examples of Berkovich Julia sets that are connected but are not contained in a line segment.The main engine we use to produce our examples is the following theorem. To state it, we note that a finite tree Γ ⊆ P 1Ber is exactly what it sounds like: a subset of P 1 Ber homeomorphic
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