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  • Print publication year: 2006
  • Online publication date: February 2010

6 - The ergodic partition for toral linked twist maps

Summary

This chapter discusses the application of Pesin theory to linked twist maps. Drawing on three key papers from the ergodic theory literature we give the proof that linked twist maps can be decomposed into at most a countable number of ergodic components.

Introduction

In Chapter 4 we gave Devaney's construction of a horseshoe for a linked twist map on the plane. The existence of the horseshoe and the accompanying subshift of finite type implies that the linked twist map contains a certain amount of complexity. However, topological features such as horseshoes may not be of interest from a statistical, observable, or measure-theoretic point of view, as they occur on invariant sets of measure zero. The subshift of finite type occurs on just such an invariant set of measure zero and is therefore arguably not of significant statistical interest. Nevertheless it is possible that similar behaviour is shared by points in the vicinity of the horseshoe, meaning that complex behaviour is present in a significant (that is, positive measure) domain. Easton (1978) conjectures that this may indeed be the case, and that in fact linked twist maps may be ergodic.

Three papers provide the framework for applying the results of Pesin (1977) connecting hyperbolicity and ergodicity. In this and the following two chapters we draw heavily on each of Burton & Easton (1980), Wojtkowski (1980) and Przytycki (1983).

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