Thurston Maps

Abstract

In this chapter, we introduce the dynamical systems that we are going to study, namely, expanding Thurston maps. We first recall briefly some history in Sect. 2.1, where we by no means intend to give a complete account of the development of the subject. We then introduce Thurston maps in Sect. 2.2 and certain cell decompositions of the 2-sphere $S^2$ induced by Thurston maps in Sect. 2.3. Next, we discuss various notions of expansion in our context and define expanding Thurston maps in Sect. 2.4. Two most important tools in the study of expanding Thurston maps are explored in the last two sections. The first tool is a natural class of metrics on the $S^2$, called visual metrics, discussed in Sect. 2.5. The second tool, discussed in Sect. 2.6, is the existence and properties of certain forward invariant Jordan curves on $S^2$, which induce nice partitions of the sphere. It is the geometric and combinatorial information we get from these tools that enables us to investigate the dynamical properties of expanding Thurston maps.