# Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks

@article{McMullen2002DynamicsOK, title={Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks}, author={Curtis T. McMullen}, journal={Crelle's Journal}, year={2002}, volume={2002}, pages={201-233} }

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…

## 161 Citations

K3 Surfaces, Picard Numbers and Siegel Disks

- Mathematics
- 2021

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct…

DYNAMICS ON SUPERSINGULAR K3 SURFACES AND AUTOMORPHISMS OF SALEM DEGREE 22

- MathematicsNagoya Mathematical Journal
- 2016

In this paper, we exhibit explicit automorphisms of maximal Salem degree 22 on the supersingular K3 surface of Artin invariant one for all primes $p\equiv 3~\text{mod}\,4$ in a systematic way.…

The third smallest Salem number in automorphisms of K3 surfaces

- Mathematics
- 2009

We realize the logarithm of the third smallest known Salem number as the topological entropy of a K3 surface automorphism with a Siegel disk and a pointwisely fixed curve at the same time. We also…

Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups

- Mathematics
- 2011

Thanks to the theory of Coxeter groups, we produce the first family of Calabi-Yau manifolds $X$ of arbitrary dimension $n$, for which ${\rm Bir}(X)$ is infinite and the Kawamata-Morrison movable cone…

Automorphisms of minimal entropy on supersingular K3 surfaces

- MathematicsJ. Lond. Math. Soc.
- 2018

A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, i.e. preserves a chamber of the positive cone.

Automorphism groups and anti-pluricanonical curves

- Mathematics
- 2007

We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z . Z (semi-direct…

Elliptic fibrations on K3 surfaces and Salem numbers of maximal degree

- MathematicsJournal of the Mathematical Society of Japan
- 2018

We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. By generalizing \cite{EOY14}, we establish a characterization of such maximum in terms of elliptic…

Automorphismes d'entropie positive, le cas des surfaces rationnelles

- Mathematics
- 2010

A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be a torus, a K3 surface, an Enriques surface or a non-minimal rational surface.…

Automorphisms of Supersingular K3 Surfaces and Salem Polynomials

- MathematicsExp. Math.
- 2016

It is shown that if p is an odd prime less than or equal to 7919, then every supersingular K3 surface in characteristic p has an automorphism whose characteristic polynomial on the Néron–Severi lattice is a SalemPolynomial of degree 22.

Finite orbits for large groups of automorphisms of projective surfaces

- Mathematics
- 2020

We study finite orbits for non-elementary groups of automorphisms of compact projective surfaces. In particular we prove that if the surface and the group are defined over a number field k and the…

## References

SHOWING 1-10 OF 70 REFERENCES

Rational points on K3 surfaces: A new canonical height

- Mathematics
- 1991

A fundamental tenet of Diophantine Geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties. This is certainly true for curves, where the…

Anosov mapping class actions on the $SU(2)$-representation variety of a punctured torus

- MathematicsErgodic Theory and Dynamical Systems
- 1998

Recently, Goldman [2] proved that the mapping class group of a compact surface $S$, ${\it MCG}(S)$, acts ergodically on each symplectic stratum of the Poisson moduli space of flat $ S(2)$-bundles…

On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

- Mathematics
- 1978

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the…

Complex Dynamics and Renormalization

- Mathematics
- 1994

Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid…

Compact complex surfaces.

- Mathematics
- 2003

Historical Note.- References.- The Content of the Book.- Standard Notations.- I. Preliminaries.- Topology and Algebra.- 1. Notations and Basic Facts.- 2. Some Properties of Bilinear forms.- 3. Vector…

Dynamics in One Complex Variable: Introductory Lectures

- Mathematics
- 2000

These notes study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on…

Entropy of Algebraic Maps

- Mathematics
- 1995

In this paper I give upper bounds for the entropy of algebraic maps in terms of certain homological data induced by their graphs. AMS classification 28D20, 30D05, 54H20 §0. Introduction Let X be a…

Salem numbers of negative trace

- MathematicsMath. Comput.
- 2000

We prove that, for all d > 4, there are Salem numbers of degree 2d and trace - 1, and that the number of such Salem numbers is » d/ (log log d) 2 . As a consequence, it follows that the number of…

Symmetric Bilinear Forms

- Mathematics
- 1973

I. Basic Concepts.- II. Symmetric Inner Product Spaces over Z.- III. Inner Product Spaces over a Field.- IV. Discrete Valuations and Dedekind Domains.- V. Some Examples.- Appendix 1. Quadratic…

Volume growth and entropy

- Mathematics
- 1987

An inequality is proved, bounding the growth rates of the volumes of iterates of smooth submanifolds in terms of the topological entropy. ForCx-smooth mappings this inequality implies the entropy…