Weak Mixing for Interval Exchange Transformations and Translation Flows

Introduction to Teichmüller Theory and Its Applications to Dynamics of Interval Exchange Transformations, Flows on Surfaces and Billiards

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school “Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory” (from 4 to 16 July, 2011). In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teich...

Weak mixing of maps with bounded cutting parameter

In the class of Ornstein transformations the mixing property satisfies a 0-1 law. Here we consider Ornstein’s construction with bounded cutting parameter. In fact, these latter transformations are not mixing, however it is proved that the weak mixing property occurs with probability one. Our situation is similar to the case of interval exchange transformations whose link with the cutting and st...

Topological Dynamics

1. Dynamical embedding 1.1. Uniformly distributed sequence generators 1.2. Weak mixing relative to a partition 1.3. Construction of P-mixing transformations 1.4. Well distributed sequences 1.5. A theorem in Diophantine approximation 1.6. Disjointness in topological dynamics 1.7. Proof of Theorem 1.5.1 1.8. Quasiregular points and generic points 1.9. Normal sequences and collectives 1.10. Disjoi...

Structure of Three-interval Exchange Transformations Iii: Ergodic and Spectral Properties

In this paper we present a detailed study of the spectral/ergodic properties of threeinterval exchange transformations. Our approach is mostly combinatorial, and relies on the diophantine results in Part I and the combinatorial description in Part II. We define a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measuretheoretic point of vi...

Dynamics of Interval Exchange Transformations and Teichmüller Flows