Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks

By Curtis T. McMullen. 2002

198 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/89107ab74d06b67b3fb49691a6a3363a65157aa0

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Citations in the corpus

Seventy years of Salem numbers (2014, unknown): “This is because McMullen’s proof for K3 surfaces in [81] is readily generalized.”
Distributions in algebraic dynamics (2005, methodology): “For classification and complex theory of automorphisms of surfaces, in particular K3 surfaces, we refer to Cantat [15, 14] and McMullen [40, 39].”
Lectures on K3 Surfaces (2016, result): “This started with the two articles by Cantat and McMullen [100, 401], but see also [472] for recent progress and references.”
Dynamics of automorphisms on projective complex manifolds (2008, methodology): “Since L is an integral lattice of signature (1, r) and g∗|L is an isometry of L, by the proof of McMullen [36] Theorem 3.2, d1 is a Salem number and all eigenvalues of g ∗|L are given as: d1, d −1 1 , α1, ᾱ1, . . . , αt, ᾱt with |αi| = 1 and 2(t + 1) = r + 1.”
Suites d’Applications Méromorphes Multivaluées et Courants Laminaires (2005, background): “Notons que dans le cas des automorphismes du type Hénon, la laminarité de courants de Green a permis de démontrer de nombreuses propriétés dynamiques importantes (voir aussi [5, 7, 22, 14] pour le cas des applications biméromorphes sur les surfaces).”
Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy (2013, background): “neralProblem. Complex dynamicsof biholomorphicautomorphismsofcompact Ka¨hler surfaces is now fairly well-understood since Cantat [Ca99]. Especially, Bedford-Kim ([BK09], [BK12], [BK10]) and McMullen ([Mc02], [Mc07], [Mc11], [Mc13]) show very beautiful aspects of automorphisms of rational surfaces an”
Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences (2006, background): “We show in §4 that this gives a counterexample of a conjecture/question of Gizatullin, Harbourne, and McMullen.”
Automorphisms of even unimodular lattices and unramified Salem numbers (2002, background): “Our study of automorphisms of lattices began with K3 surfaces [ Mc ], and we conclude with an application to these varieties.”
Groupes commutatifs d'automorphismes d'une variété kählérienne compacte (2002, background): “Le lecteur peut consulter les articles de Cantat [4], Mazur [17] et McMullen [18] pour une liste d’exemples et une étude dynamique des automorphismes d’une surface complexe.”
Dynamics on blowups of the projective plane (2007, methodology): “Siegel disk. A Siegel disk is possible on a K3 surface, but only when S is nonprojective [ Mc2 ].”
Non-K\"ahler Calabi-Yau manifolds (2014, background): “There are also examples with X a K3 surface, due to McMullen [58], and an explicit example with X a 3-dimensional complex torus, due to Iitaka [79, Remark 14.”
Green currents for holomorphic automorphisms of compact Kähler manifolds (2004, background): “In dimension 2, many examples are given in [33, 6, 34].”
A note on automorphisms and birational transformations of holomorphic symplectic manifolds (2009, background): “ complete understanding of the ﬁnite abelian automorphism groups of K3 surfaces. These results have been extended to non abelian groups by Mukai [23], and to inﬁ- nite order automorphisms by McMullen [22]. In higher dimension, Debarre [14] and Namikawa [24] provided counter-examples to various analo”
Limits of Calabi–Yau metrics when the Kähler class degenerates (2007, background): “ults of Section 8 in [Fi], where the equation of McLean’s metric appears in his condition (C). The situation is diﬀerent when αis not c 1(L), and Xpossibly not projective. Then an example of McMullen [McM] shows that the Ricci-ﬂat metrics can converge smoothly to zero on an open set of X. Also easy ”
Non-Kähler Calabi-Yau manifolds (2014, background): “There are also examples with X a K3 surface, due to McMullen [58], and an explicit example with X a 3-dimensional complex torus, due to Iitaka [79, Remark 14.”