Pseudoautomorphisms with Invariant Curves

Abstract

Nontrivial automorphisms of complex compact manifolds are typically rare and more typically non-existent. It is interesting to understand which manifolds admit automorphisms, how plentiful they are on any given manifold, and what further special properties distinguish a particular automorphism, or family of automorphisms. These problems have enjoyed much attention in the past fifteen years, motivated largely by work in complex dynamics (e.g. Cantat's thesis [7]). In this introduction, we give a quick account of some of this research, introducing in particular the more general category of pseudoautomorphisms, which occur more frequently in higher dimensions than automorphisms. The final aim of our paper is to present a concrete alternative approach to some recent existence results [22, Theorems 1.1 and 3.1] of Perroni and Zhang for pseudoautomorphisms with invariant elliptic curves on rational complex manifolds. Our methods lead to explicit formulas which are especially simple (see Theorems 4.7 and 6.4) when the pseudoautomorphisms correspond to the 'Coxeter element' in an infinite, finitely generated reflection group.The topological entropy of an automorphism is a non-negative number that measures the complexity of point orbits. 'Positive entropy' will serve as a precise and reasonable necessary condition for a map to be dynamically interesting. In complex dimension one, i.e. on closed Riemann surfaces, there are no automorphisms of positive entropy. In dimension two, Cantat [7] showed that only three types of complex surfaces can carry automorphisms of positive entropy: tori, K3 surfaces (or certain quotients), or rational surfaces. Automorphisms of tori are essentially linear. The cases of K3 and rational surfaces are much more interesting. Dynamics of automorphisms of K3 surfaces were studied in detail by Cantat [8]. McMullen [17] constructed examples which exhibit rotation domains (two dimensional 'Siegel disks'). The family of all K3 surfaces has dimension 20, and the maximum dimension of a continuous family of K3 surface automorphisms is even smaller. By contrast, there are continuous families of rational surface automorphisms which have arbitrarily large dimension [5].It is known [20,12] that rational complex surfaces X that carry automorphisms of positive entropy are in fact modifications (i.e. compositions of point blowups) π : X → P 2 of the complex projective plane P 2 . Thus a rational surface automorphism F X : X → X with positive entropy descends via π to a birational 'map' F : P 2 P 2 which is locally biholomorphic at generic points but also has a finite union of exceptional curves that are contracted to points and conversely a finite collection I(F ) of indeterminate points which are (in a precise sense) each mapped to an algebraic curve. Since the group of all birational maps F : P 2 P 2 is quite large, this suggests trying to find automorphisms by looking at a promising family of plane birational maps and identifying those elements whose exceptional/indeterminate behavior can be eliminated by ...

Entropy of real rational surface automorphisms: Actions on the Fundamental Groups

“…Let f : P 2 (C) P 2 (C) be a quadratic birational map of the form f = T − • J • (T + ) −1 where J[x 1 : x 2 : x 3 ] = [x 2 x 3 : x 1 x 3 : x 1 x 2 ] is the Cremona involution and T ± ∈ Aut(P 2 (C)). This family of birational maps has been studied by several authors [2,4,14,21,22]. In this section, we introduce the notation and basic properties needed for the following sections.…”

“…For the real diffeomorphism associated with the orbit data 1, 1, n with a cyclic permutation, the induced action (1,7), (1,10), (2,8), (2,10), (3,2), (3,8), (3,9), (4, 10), (5, 2), (5,8), (5,9), (6, 1), (6, 5), (7, 2), (8, 1), (9, 1), (9, 2), (10, 3), (10,5), (10,6)}.…”

“…For a real diffeomorphism f R associated with the orbit data 1, 4, 5 and a cyclic permutation, the topological entropy of f : X(C) → X(C) is the logarithm of the largest real root λ ≈ 1.29349 of χ(t) = (t − 1)(t 10 − t 8 − t 7 + t 5 − t 3 − t 2 + 1) given in (1). On the other hand, using the formula (2) we see that the exponential growth rate of the induced action on the homology classes is given by the largest absolute value of the roots ≈ 1.12571 of the roots of a symmetric polynomial φ(t) = t 10 + t 8 + t 7 + t 5 + t 3 + t 2 + 1. Applying Theorem 4.7 together with Lemma 4.6 using Γ, ∆, S, M and v λ in Appendix B.3, we conclude that the growth rate of the induced action on the fundamental group ρ(f…”

Pseudo-automorphisms with no invariant foliation

“…See also [2] for related constructions of pseudo-automorphisms. The condition for f to be a pseudo-automorphism is, loosely speaking, the condition that the forward orbit of each Σ k lands on one of the points of indeterminacy e j .…”

Some dynamical properties of pseudo-automorphisms in dimension 3

“…The author is thankful to Eric Bedford for his kindly informing the results in [6] [7][8] and for useful comments on a first version of this paper. He would like to thank Tien-Cuong Dinh and Viet-Anh Nguyen for some helpful conversations on this topic.…”