Automorphisms of rational surfaces with positive entropy

Abstract

Abstract. -A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be either a torus, a K3 surface, an Enriques surface or a rational surface. Automorphisms of rational surfaces are quite mysterious and have been recently the object of intensive studies. In this paper, we construct several new examples of automorphisms of rational surfaces with positive topological entropy. We also explain how to define and to count parameters in families of birational maps of P 2 (C) and in families of rational surfaces.

Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces

“…ii) It also holds for other special cubics maps: some with two proper base-points (see [BedDil06]), with one proper base-point (see [BedKim10]), or some maps of degree 3 with exactly two proper base-points (see [BedDil06] and [DésGri10]). …”

“…This approach gives rise to the following question of Eric Bedford, also stated and studied by Julie Déserti and Julien Grivaux in [DésGri10]: Question 1.5. Does there exist a birational map of the projective plane ϕ of degree > 1 such that for all τ ∈ Aut(P 2 ) the map τ ϕ is not birationally conjugate to an automorphism of dynamical degree > 1?…”

Infinitesimal deformations of rational surface automorphisms

“…Biregular automorphisms of rational surfaces with positive topological entropy present a major interest in complex dynamics (see the recent survey [12]) but their construction remains still a difficult problem of algebraic geometry. For an overview of this problem, we refer to [13] and to the references therein. It is a paradoxal fact that these automorphisms, although hard to construct, can occur in holomorphic families of arbitrary large dimension, as shown recently in [4].…”

“…Besides, the automorphism group of a given rational surface can carry many automorphisms of positive entropy, see [5] and [6] for recent results on this topic. In this paper we study deformations of families of rational surface automorphisms using deformation theory (this was initiated in [13]), and investigate a great number of examples.…”

Automorphisms of blowups of threefolds being Fano or having Picard number 1

“…While there are many examples of compact complex surfaces having automorphisms of positive entropies (works of Cantat [10], Bedford-Kim [5][6] [7], McMullen [27][28] [29][30], Oguiso [32][33], Cantat-Dolgachev [11], Zhang [46], Diller [17], Déserti-Grivaux [16], Reschke [38],...), there are few interesting examples of manifolds of higher dimensions having automorphisms of positive entropies (Oguiso [34][35], Oguiso-Perroni [31],...). In particular, for the class of smooth rational threefolds, there are currently only two known examples of manifolds with primitive automorphisms of positive entropy (see [36,13,12]).…”