and one repelling, and the homeomorphism behaves similarly to a hyperbolic automorphism of the Poincaré half-plane. A "generic" diffeomorphism of a surface of genus at least two is isotopic to a pseudo-Anosov diffeomorphism. Generalization Using the theory of train tracks, the notion of a pseudo-Anosov map has been extended to self-maps of graphs (on the topological side) and outer automorphisms of free groups (on the algebraic side). This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by Bestvina and Handel. References A. Casson, S. Bleiler, "Automorphisms of Surfaces after Nielsen and Thurston", (London Mathematical Society Student Texts 9), (1988). A. Fathi, F. Laudenbach, and V. Poénaru, "Travaux de Thurston sur les surfaces," Asterisque, Vols. 66 and 67 (1979). R. C. Penner. "A construction of pseudo-Anosov homeomorphisms", Trans. Amer. Math. Soc., 310 (1988) No 1, 179–197 Dynamical systems Geometric topology Homeomorphisms