Abstract

An affine surface S_0 (over an algebraically closed field K) is a subset of K^n of dimension 2 given by polynomial equations. A endomorphism of S_0 is then a polynomial transformation of K^N that preserves S_0. We are interested in studying the dynamics of such a transformation. The first example of an affine surface is the complex affine plane. To study the dynamics of polynomial endomorphism of C^2, Favre and Jonsson used valuative techniques. Namely, they show that every polynomial endomorphism f admits an attracting fixed point in a certain valuation space associated to C^2. This has a strong impact on the dynamics of f as they use it to show that there exists a "nice" compactification of C^2 where f admits an attracting fixed point at infinity. With these techniques they show the following result: the dynamical degree of f (i.e the asymptotic growth rate of the degrees of the iterates of f) is an algebraic integer of degree at most 2.

I extended these techniques to every affine surface over any algebraically closed field, in particular the result on dynamical degrees holds for any affine surface over any field. In this mini-course I will introduce valuations and the valuative techniques used by Favre and Jonsson and how they apply for any affine surface. We will show the existence of a "nice" compactification where the dynamics is well understood and derive several results on the dynamics of endomorphisms of affine surfaces. In particular, in the case of an automorphism, we will see that these valuative techniques allow one to give a very precise description of the dynamics. Finally, we will study examples of affine surfaces other than the affine plane.

I will do two more lectures for my mini course. We will describe the valuative tree constructed by Favre and Jonsson and prove the main result on dynamical degrees of endomorphisms of affine surfaces.

Sources

9:00-10:30am, October 26，November 2, 9, 16, 23, and December 7, 2023