# On the dynamics of endomorphisms of affine surfaces

- Authors
- Publication Date
- Jul 07, 2023
- Source
- HAL-Descartes
- Keywords
- Language
- English
- License
- Unknown
- External links

## Abstract

An affine surface (over an algebraically closed field) is a variety of dimension 2 defined by polynomial equations. Given an endomorphism of such a surface, we can ask the following questions: Are there Zariski dense orbits ? If the orbit of a point goes to infinity, can we control the speed of divergence ? Are there a lot of periodic orbits ? Can we exhibit an interesting probability measure invariant by the dynamical system ? To answer these questions, I use valuative techniques. More precisely, the endomorphism f of an affine surface S induces a self transformation of the space of valuations centered at infinity of S. The study of the dynamics of f over this space of valuations allows one to understand the dynamics of f over S.