Abstract
This paper presents the first examples of K3 surface automorphisms \(f : X \rightarrow X\) with Siegel disks (domains on which f acts by an irrational rotation). The set of such examples is countable, and the surface \(X\) must be non-projective to carry a Siegel disk. These automorphisms are synthesized from Salem numbers of degree 22 and trace −1, which play the role of the leading eigenvalue for \(f*|H^2(X)\). The construction uses the Torelli theorem, the Atiyah-Bott fixed-point theorem and results from transcendence theory.
Highlights
The first dynamically interesting automorphisms of compact complex manifolds arise on K3 surfaces
Every such automorphism has positive topological entropy. These Siegel disks are invisible to us: they live on nonprojective K3 surfaces, and we can only detect them implicitly, through Hodge theory and dynamics on the cohomology
Theorem 3.5 If f is an automorphism of a projective K3 surface X, δ(f ) is a root of unity
Summary
The first dynamically interesting automorphisms of compact complex manifolds arise on K3 surfaces. The intermediate parameter A = 2.5 exhibits a mixture of behaviors: elliptic islands seem to coexist with an ergodic component of positive measure The dynamics in these real examples is typical for area-preserving maps on surfaces. Every such automorphism has positive topological entropy. These Siegel disks are invisible to us: they live on nonprojective K3 surfaces, and we can only detect them implicitly, through Hodge theory and dynamics on the cohomology. Theorem 1.3 Let f : X → X be a K3 surface automorphism with a Siegel disk U.
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