Indecomposable continua and exotic rotation domains

Abstract

This thesis studies the construction of dynamical systems exhibiting exotic geometric and topological behavior. A central role is played by the approximation-by-conjugation method introduced by Anosov and Katok, which produces smooth dynamical systems as limits of conjugates of simple rotations and allows for the creation of examples with rich ergodic and geometric properties.

We present in detail Herman’s construction of smooth pseudo-rotations of the sphere with an invariant pseudo-circle, a hereditarily indecomposable continuum with highly intricate topology. This construction shows how dynamical systems that are close to rigid rotations can nevertheless generate extremely complicated invariant sets. We also present a detailed construction of the pseudo-circle.

In addition, we consider Chéritat’s work on holomorphic maps with Siegel disks whose boundaries are nowhere locally connected, even when the rotation domain is compactly contained in the domain of definition. Unlike Herman’s examples, it does not rely on the Anosov–Katok method. We present our own approach to Ch´eritat’s theorem, a construction designed to be as transparent as possible, with the goal of making explicit the arithmetic properties of the rotation numbers that allow this phenomenon to occur.

The thesis also includes auxiliary results on the gluing of diffeomorphisms required in Herman’s construction, together with some general discussions on variations of the constructions. We emphasize the interplay between arithmetic properties of rotation numbers, approximation techniques, and the geometry of invariant sets.