Summer School of Mathematics "Berkovich Spaces"
Dynamics on Berkovich Spaces (Lecture 3) Mattias Jonsson (Univ. Michigan, IHES) 30 juin 2010 
In these lectures I will present two instances where dynamics on Berkovich spaces appear naturally.
• The first case is in the context of iterations of selfmaps of the (standard) projective line over a nonArchimedean field such as the padic numbers. When trying to extend results from the Archimedean setting (over the complex numbers), it turns out be both natural and fruitful to study the induced dynamics on the associated Berkovich projective line.
• The second case concerns iterations of (germs of) holomorphic selfmaps f:C2>C2 fixing the origin, f(0)=0. When the fixed point is superattracting, that is, the differential df(0) is identically zero, the dynamics can be analyzed by studying the induced action on the Berkovich affine plane over the field C equipped with the trivial valuation.
Beyond the subject of dynamics, these lectures will provide a "handson" introduction to Berkovich spaces in relatively concrete settings, where the topological structure is essentially that of an Rtree. In studying the second instance above, we will also have the opportunity to explore the link between Berkovich spaces and the algebrogeometric study of valuations, going back to Zariski. If time permits, I will also other dynamical situations, such as the behavior at infinity of iterates of twodimensional polynomial selfmaps. I may also briefly discuss the higherdimensional case.
– Lecture 1: June 28, 15:30
– Lecture 2: June 29, 10:30
– Lecture 3: June 30, 10:30
– Lecture 4: July 1, 09:00
– Lecture 5: July 2, 11:30
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Mattias Jonsson (Univ. Michigan, IHES) Mattias Jonsson is professor of mathematics at the University of Michigan. 