Abstract:We consider the collection of all pseudo-Anosov homeomorphisms on a surface of fixed topological type. To each such homeomorphism is associated a real-valued invariant, called its dilatation (which is greater than one), and we define the spectrum of the surface to be the collection of logarithms of dilatations of pseudo-Anosov maps supported on the surface. The spectrum is a natural object of study from the topological, geometric, and dynamical points of view. We are concerned in this paper with the least element of the spectrum, and explicit upper and lower bounds on this least element are derived in terms of the topological type of the surface; train tracks are the main tool used in establishing our estimates.
- William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. MR 590044
- Pierre Arnoux and Jean-Christophe Yoccoz, Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 75–78 (French, with English summary). MR 610152 M. Bauer, Examples of pseudo-Anosov homeomorphisms, Thesis, Univ. of Southern California, 1989.
- Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308 F. Gantmacher, Theory of matrices, vol. 2, Chelsea, 1960. A. Papadopoulos, Reseaux Ferroviares, diffeomorphismes pseudo-Anosov et Automorphismes symplectiques de l’homologie, Publ. Math. d’Orsay 83-103 (1983).
- Robert C. Penner, A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988), no. 1, 179–197. MR 930079, DOI 10.1090/S0002-9947-1988-0930079-9
- R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770, DOI 10.1515/9781400882458
- Athanase Papadopoulos and Robert C. Penner, A characterization of pseudo-Anosov foliations, Pacific J. Math. 130 (1987), no. 2, 359–377. MR 914107 W. Thurston, The geometry and topology of three-manifolds, Ann. Math. Stud., Princeton Univ. Press, Princeton, NJ (to appear).
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 443-450
- MSC: Primary 57N05; Secondary 32G15, 57M20, 58F18
- DOI: https://doi.org/10.1090/S0002-9939-1991-1068128-8
- MathSciNet review: 1068128