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Equivariant Linearization and Rotation Domains on K3 Surfaces

Katsunori Iwasaki

TL;DR

The paper develops a comprehensive framework to construct and analyze K3 surface automorphisms of positive entropy that exhibit rotation domains of rank 1 and 2. It fuses equivariant linearization near exceptional components with Lefschetz-type fixed-point formulas and the method of hypergeometric groups to produce and study explicit examples across Dynkin types A, D, and E, often with intricate symmetry properties. A core achievement is a detailed semi-local to global toolkit: linear models near exceptional sets capture local dynamics, while global arithmetic (Salem numbers and MDR criteria) governs multipliers at periodic points, enabling rigorous classification of rotation domains and their centers. The results yield a rich inventory of rotation-domain phenomena, including coexistence of ranks, and periodic cycles with centers or hyperbolic behavior, advancing both the construction and the dynamics understanding of K3 automorphisms with positive entropy. The work provides practical procedures to detect rotation domains from fixed-point data and demonstrates the power of combining geometric resolution theory with hypergeometric-group methods for explicit dynamical systems on K3 surfaces.

Abstract

We construct a lot of K3 surface automorphisms of positive entropy having rotation domains of ranks 1 and 2. To carry out this construction, we first lay theoretical foundations concerning equivariant linearization of nonlinear maps under resolutions of quotient singularities, linear models near exceptional components, Salem numbers and multipliers at periodic points, two kinds of fixed point formulas and related indices at exceptional components. Then these basic tools are combined with the method of hypergeometric groups to enable us to detect various types of rotation domains on K3 surfaces.

Equivariant Linearization and Rotation Domains on K3 Surfaces

TL;DR

The paper develops a comprehensive framework to construct and analyze K3 surface automorphisms of positive entropy that exhibit rotation domains of rank 1 and 2. It fuses equivariant linearization near exceptional components with Lefschetz-type fixed-point formulas and the method of hypergeometric groups to produce and study explicit examples across Dynkin types A, D, and E, often with intricate symmetry properties. A core achievement is a detailed semi-local to global toolkit: linear models near exceptional sets capture local dynamics, while global arithmetic (Salem numbers and MDR criteria) governs multipliers at periodic points, enabling rigorous classification of rotation domains and their centers. The results yield a rich inventory of rotation-domain phenomena, including coexistence of ranks, and periodic cycles with centers or hyperbolic behavior, advancing both the construction and the dynamics understanding of K3 automorphisms with positive entropy. The work provides practical procedures to detect rotation domains from fixed-point data and demonstrates the power of combining geometric resolution theory with hypergeometric-group methods for explicit dynamical systems on K3 surfaces.

Abstract

We construct a lot of K3 surface automorphisms of positive entropy having rotation domains of ranks 1 and 2. To carry out this construction, we first lay theoretical foundations concerning equivariant linearization of nonlinear maps under resolutions of quotient singularities, linear models near exceptional components, Salem numbers and multipliers at periodic points, two kinds of fixed point formulas and related indices at exceptional components. Then these basic tools are combined with the method of hypergeometric groups to enable us to detect various types of rotation domains on K3 surfaces.
Paper Structure (27 sections, 55 theorems, 177 equations, 3 figures, 3 tables)

This paper contains 27 sections, 55 theorems, 177 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Equivariant Linearization
  3. Resolutions of Singularities
  4. Cyclic Cases
  5. Non-Cyclic Cases
  6. Normalizers on Minimal Resolutions
  7. Cyclic Cases
  8. Non-Cyclic Cases
  9. Linear Models near Exceptional Components
  10. Salem Numbers and Multipliers at Periodic Points
  11. Non-Resonance
  12. Some Properties of Salem Numbers
  13. Multiplicative Dependence and Non-Resonance
  14. Fixed Point Formulas
  15. Reformulations of Fixed Point Formulas
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $E$ be an $f$-invariant exceptional component. If $E$ is of Dynkin type $\mathrm{D}$ or $\mathrm{E}$, then $(\mathrm{R} 1)$ is always the case for $(f, E)$. The same is true when $E$ is of Dynkin type $\mathrm{A}$ with reflectional symmetry. On the other hand, when $E$ is of Dynkin type $\mathrm

Figures (3)

  • Figure 3.1: Dynkin diagram of type $\mathrm{D}$ or $\mathrm{E}$ and its arms.
  • Figure 4.1: The case of $n = 2 k$ with $k \ge 1$.
  • Figure 4.2: The case of $n = 2 k-1$ with $k \ge 2$.

Theorems & Definitions (75)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 65 more