Hölder Continuous Paths and Hyperbolic Automorphisms of T3

Abstract

Publisher Summary This chapter focuses on Holder continuous paths and hyperbolic automorphisms of T 3 . A hyperbolic toral automorphism f : T n → T n is a diffeomorphism of the torus that lifts to a hyperbolic automorphism of R n . A subset K of T n is invariant if it is compact and f ( K ) = K . On iteration, f mixes up the torus very thoroughly, and this puts heavy restrictions on invariant sets K . The chapter is concerned with two questions: what dimensions K can have and how smooth it can be. By dimension, it means, in the first instance, topological (covering, inductive) dimension, which is an integer. Some examples of invariant sets K are given in the chapter. There are various theorems about invariant sets K of a hyperbolic toral automorphism f : T n → T n .. Three results that are particularly relevant are also presented in the chapter.