G\'eom\'etrie et dynamique sur les surfaces alg\'ebriques r\'eelles [Geometry and dynamics on real algebraic surfaces]

TL;DR

This thesis investigates the dynamics of automorphisms on real algebraic surfaces, linking real and complex entropy through a geometric quantity called concordance, and analyzing the Fatou set's properties to understand real dynamics.

Contribution

It introduces the concept of concordance to relate real and complex entropy and computes it for various surfaces, advancing understanding of real dynamics on algebraic surfaces.

Findings

The ratio of real to complex entropy is linked to the concordance.

The Fatou set is hyperbolic in the Kobayashi sense after removing certain curves.

Most real loci exhibit complex dynamics, not entirely contained in the Fatou set.

Abstract

This thesis deals with automorphisms of real algebraic surfaces, which are polynomial transformations with a polynomial inverse. The main concern is whether their restriction to the real locus reflects all the richness of the complex dynamics. This question is declined in two directions: the topological entropy and the Fatou set. For the first one, we introduce a purely geometric quantity depending only on the surface, and we call it concordance. Then we show that the ratio of real and complex entropies is linked to this quantity. The concordance is explicitely computed for many examples of surfaces, especially abelian surfaces which are broadly studied, as well as some K3 surfaces. In the second part, we are interested in the Fatou set, which corresponds to complex points for which the dynamics is simple. Thanks to previous results of Dinh and Sibony about closed positive currents, we prove that this set is hyperbolic in the sense of Kobayashi, after possibly deleting some curves which are fixed by (an iterate of) our transformation. From this property we deduce that, except for some exceptional cases in which the topology of the real locus is simple and the dynamics well understood, this real locus cannot be entirely contained in the Fatou set. Thus the complexity of the dynamics is observable on real points in most cases.