370 13.2 THE THREE TYPES OF MAPPING CLASSES CHAPTER 13 We now describe three kinds of elements of the mapping class group of a surface. Each is an analogue of one of the three types of elements of Mod(T 2 ) in the statement of Theorem 13.1. The Nielsen­Thurston clas- sification theorem (Section 13.3) states that every mapping class falls into (at least) one of these three categories. 13.2.1 P ERIODIC M APPING C LASSES We have already studied periodic, or finite-order, elements of the mapping class group. A basic example is shown in Figure 13.1; see also Figures 2.1­ 2.3. See Chapter 7 for a general discussion of finite-order mapping classes.