On the dynamics of birational diffeomorphism of real algebraic surfaces: Fatou sets and the real line (Q355225)

The paper investigates dynamics in the situation where a birational self-map on a complex manifold restricts to a self-diffeomorphism on the real locus of the manifold. Examples of this situation where the birational map is an automorphism are abundant, and they include instances in which the real locus contains all of the interesting dynamics of the automorphism. However, the main result of the paper is a construction of a birational map with complicated dynamics that are invisible on the real locus (even though, importantly, the diffeomorphism on the real locus determines the birational map).NEWLINENEWLINEThe main example in the paper is a birational self-map on \(X = \mathbb{P}^1 \times \mathbb{P}^1\). The (fairly strong) condition on the real dynamics obtained in the example is that \(X(\mathbb{R})\) is contained in the Fatou domain of the birational map; indeed, there is a neighborhood (which is a product of two complex annuli) of \(X(\mathbb{R})\) (which is a real two-torus) in \(X\) on which the birational map is biholomorphic and conjugate to a rotation. The assertion that the birational map has complicated dynamics on \(X\) away from \(X(\mathbb{R})\) is encompassed by the condition that the dynamical degree of the map is greater than one. Note that the construction in the paper is not quite explicit, as it depends on an existence statement about certain rotations in the diffeomorphism group of a real two-torus.NEWLINENEWLINEThe paper concludes by showing that any compact complex surface admitting a birational self-map as in the main example must have a real locus whose components are only spheres, tori, projective planes, or Klein bottles. It remains unknown if examples can be constructed where the real locus is not a torus or if examples can be constructed where the birational map is actually biholomorphic.