Title: On the rigidity of periodic points for automorphisms of affine surfaces
 

Abstract: I will discuss the following results. Let S be a complex affine surface and f,g be two automorphisms of positive entropy. If f and g have a Zariski dense set of periodic points in common then they have the same set of periodic points. The proof uses the dynamics at infinity of such automorphisms and the construction of their canonical Green functions and equilibrium measures both for archimedean places and non-archimedean ones. One of the main ingredients is the theorem of arithmetic equidistribution on adelic line bundles over quasiprojective varieties from Yuan and Zhang. I will also discuss examples of affine surfaces where I manage to show a stronger rigidity: having the same periodic points imply that the automorphisms share a common iterate. The examples are the affine plane and Markov surfaces which are related to the character variety of the punctured torus.