In this note we characterize the itineraries associated to Lorenz maps and the conjugacy classes for this class of maps.

Introduction

In this note we will discuss the topological classification problem for discontinuous piecewise monotone maps on the interval.

Definition 8.1.1 Let f : X → X and g : Y → Y be two continuous maps of topological spaces X and Y respectively. A continuous and onto map π : X → Y such that g ∘ π = π ∘ f is called a factor map and f is said to be an extension of g. We also say that g and f are semi conjugated. If π is an homeomorphism we say that f and g are topologically conjugate or topologically equivalent.

Assume that X = Y. In this situation we can ask for necessary and sufficient conditions for the maps f and g to be topologically conjugated maps. This was one of the central topics in dynamics in the last thirty years. See for instance [3], [5], [7], [4], [13] and the references therein.

In this notes we discuss these questions for the special class of Lorenz maps of the interval.