Dynamics on Berkovich Spaces in Low Dimensions Mattias Jonsson Contents 1 Introduction..................................................................................... 205 2 Tree Structures ................................................................................. 220 3 The Berkovich Affine and Projective Lines ................................................... 236 4 Action by Polynomial and Rational Maps .................................................... 253 5 Dynamics of Rational Maps in One Variable ................................................. 270 6 The Berkovich Affine Plane Over a Trivially Valued Field .................................. 283 7 The Valuative Tree at a Point .................................................................. 297 8 Local Plane Polynomial Dynamics ............................................................ 328 9 The Valuative Tree at Infinity.................................................................. 335 10 Plane Polynomial Dynamics at Infinity ....................................................... 352 References .......................................................................................... 361 1 Introduction The goal of these notes is twofold. First, I’d like to describe how Berkovich spaces enter naturally in certain instances of discrete dynamical systems. In particular, I will try to show how my own work with Charles Favre [FJ07, FJ11]onvaluative dynamics relates to the dynamics of rational maps on the Berkovich projective line as initiated by Juan Rivera-Letelier in his thesis [Riv03a] and subsequently studied by him and others. In order to keep the exposition somewhat focused, I have chosen three sample problems (Theorems A, B and C below) for which I will present reasonably complete proofs. M. Jonsson () Department of Mathematics, University of Michigan, 530 Church Street, 2076 East Hall, Ann Arbor, MI 48109-1043, USA e-mail: [email protected],Url:http://www.math.lsa.umich.edu/~mattiasj/ © Springer International Publishing Switzerland 2015 205 A. Ducros et al. (eds.), Berkovich Spaces and Applications, Lecture Notes in Mathematics 2119, DOI 10.1007/978-3-319-11029-5__6 206 M. Jonsson The second objective is to show some of the simplest Berkovich spaces “in action”. While not necessarily representative of the general situation, they have a structure that is very rich, yet can be described in detail. In particular, they are trees, or cones over trees. For the purposes of this introduction, the dynamical problems that we shall be interested in all arise from polynomial mappings f W An ! An; where An denotes affine n-space over a valued field,thatis,afieldK complete with respect a norm jj. Studying the dynamics of f means, in rather vague terms, studying the asymptotic behavior of the iterates of f : f m D f ı f ııf (the composition is taken m times) as m !1. For example, one may try to identify regular as opposed to chaotic behavior. One is also interested in invariant objects such as fixed points, invariant measures, etc. When K is the field of complex numbers, polynomial mappings can exhibit very interesting dynamics both in one and higher dimensions. We shall discuss this a little further in Sect. 1.1 below. As references we point to [CG93, Mil06]forthe one-dimensional case and [Sib99] for higher dimensions. Here we shall instead focus on the case when the norm on K is non-Archimedean in the sense that the strong triangle inequality ja C bjÄmaxfjaj; jbjg holds. Interesting examples of such fields include the p-adic numbers Qp, the field of Laurent series C..t//,oranyfieldK equipped with the trivial norm. One motivation for investigating the dynamics of polynomial mappings over non-Archimedean fields is simply to see to what extent the known results over the complex (or real) numbers continue to hold. However, non-Archimedean dynamics sometimes plays a role even when the original dynamical system is defined over the complex numbers. We shall see some instances of this phenomenon in these notes; other examples are provided by the work of Kiwi [Kiw06], Baker and DeMarco [BdM09], and Ghioca, Tucker and Zieve [GTZ08]. Over the complex numbers, many of the most powerful tools for studying dynam- ics are either topological or analytical in nature: distortion estimates, potential theory, quasiconformal mappings etc. These methods do not directly carry over to the non-Archimedean setting since K is totally disconnected. On the other hand, a polynomial mapping f automatically induces a selfmap n n f W ABerk ! ABerk n n n of the corresponding Berkovich space ABerk. By definition, ABerk D ABerk.K/ is the n set of multiplicative seminorms on the coordinate ring R ' KŒz1;:::;zn of A that extend the given norm on K. It carries a natural topology in which it locally compact and arcwise connected. It also contains a copy of An: a point x 2 An is identified Dynamics on Berkovich Spaces in Low Dimensions 207 n with the seminorm 7!j.x/j. The action of f on ABerk is given as follows. A seminorm jjis mapped by f to the seminorm whose value on a polynomial 2 R is given by jf j. n The idea is now to study the dynamics on ABerk. At this level of generality, not very much seems to be known at the time of writing (although the time may be ripe to start looking at this). Instead, the most interesting results have appeared in n situations when the structure of the space ABerk is better understood, namely in sufficiently low dimensions. We shall focus on two such situations: (1) f W A1 ! A1 is a polynomial mapping of the affine line over a general valued field K; (2) f W A2 ! A2 is a polynomial mapping of the affine plane over a field K equipped with the trivial norm. In both cases we shall mainly treat the case when K is algebraically closed. 1 In (1), one makes essential use of the fact that the Berkovich affine line ABerk is a tree.1 This tree structure was pointed out already by Berkovich in his original work [Ber90] and is described in great detail in the book [BR10] by Baker and Rumely. It has been exploited by several authors and a very nice picture of the global dynamics on this Berkovich space has taken shape. It is beyond the scope of these notes to give an account of all the results that are known. Instead, we shall focus on one specific problem: equidistribution of preimages of points. This problem, which will be discussed in further detail in Sect. 1.1, clearly shows the advantage of working on the Berkovich space as opposed to the “classical” affine line. 2 As for (2), the Berkovich affine plane ABerk is already quite a beast, but it is possible to get a handle on its structure. We shall be concerned not with the global dynamics of f , but the local dynamics either at a fixed point 0 D f.0/ 2 A2,orat 2 infinity. There are natural subspaces of ABerk consisting of seminorms that “live” at 0 or at infinity, respectively, in a sense that can be made precise. These two spaces are cones over a tree and hence reasonably tractable. While it is of general interest to study the dynamics in (2) for a general field K, there are surprising applications to complex dynamics when using K D C equipped with the trivial norm. We shall discuss this in Sects. 1.2 and 1.3 below. 1.1 Polynomial Dynamics in One Variable Our first situation is that of a polynomial mapping f W A1 ! A1 1For a precise definition of what we mean by “tree”, see Sect. 2. 208 M. Jonsson of degree d>1over a complete valued field K, that we here shall furthermore assume to be algebraically closed and, for simplicity, of characteristic zero. When K is equal to the (archimedean) field C, there is a beautiful theory describing the polynomial dynamics. The foundation of this theory was built in the 1920s by Fatou and Julia, who realized that Montel’s theorem could be used to divide the phase space A1 D A1.C/ into a region where the dynamics is tame (the Fatou set) and a region where it is chaotic (the Julia set). In the 1980s and beyond, the theory was very significantly advanced, in part because of computer technology allowing people to visualize Julia sets as fractal objects, but more importantly because of the introduction of new tools, in particular quasiconformal mappings. For further information on this we refer the reader to the books [CG93, Mil06]. In between, however, a remarkable result by Hans Brolin [Bro65] appeared in the 1960s. His result seems to have gone largely unnoticed at the time, but has been of great importance for more recent developments, especially in higher dimensions. Brolin used potential theoretic methods to study the asymptotic distribution of preimages of points. To state his result, let us introduce some terminology. Given a polynomial mapping f as above, one can consider the filled Julia set of f , consisting of all points x 2 A1 whose orbit is bounded. This is a compact set. Let f be harmonic measure on the filled Julia set, in the sense of potential theory. Now, given a point x 2 A1 we can look at the distribution of preimages of x n n under f .ThereareP d preimages of x, counted with multiplicity, and we write n n n f ıx D f nyDx ıy, where the sum is taken over these preimages. Thus d f ıx is a probability measure on A1. Brolin’s theorem now states Theorem For all points x 2 A1, with at most one exception, we have n n lim d f ıx ! f : n!1 Furthermore, a point x 2 A1 is exceptional iff there exists a global coordinate z on A1 vanishing at x such that f is given by the polynomial z 7! zd . In this case, n n d f ıx D ıx for all n. A version of this theorem for selfmaps of P1 was later proved independently by Lyubich [Lyu83] and by Freire-Lopez-Mañé [FLM83].