The theorem of Kerekjarto on periodic homeomorphisms of the disc and the sphere

TL;DR

Problem: classify periodic homeomorphisms of the disc and sphere up to topological conjugacy with Euclidean isometries. Approach: provide a modern, elementary proof using fixed-point analysis, invariant discs, Jordan–Schoenflies, and quotient/branched-covering arguments to reduce dynamics to Euclidean rotations or reflections. Contributions: explicit conjugacies to rotations or reflections for $D^{2}$ and $S^{2}$; clarification of fixed-point set structures and a unified method that recovers the classical isometric models, with a note on pointwise periodic generalizations. Significance: consolidates a foundational result in low-dimensional topology and informs related work on isometries of surfaces and higher-genus cases.

Abstract

We give a modern exposition and an elementary proof of the topological equivalence between periodic homeomorphisms of the disc and the sphere and euclidean isometries.

Figures (4)

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Theorems & Definitions (20)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3: Jordan-Schoenflies
• Proposition 2.4
• proof
• Lemma 2.5
• proof
• Remark 2.6
• Theorem 3.1
• Proposition 3.2