×

Bounds on least dilatations. (English) Zbl 0726.57013

Summary: 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.

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57N50 \(S^{n-1}\subset E^n\), Schoenflies problem
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. · Zbl 0452.32015
[2] 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). · Zbl 0478.58023
[3] M. Bauer, Examples of pseudo-Anosov homeomorphisms, Thesis, Univ. of Southern California, 1989. · Zbl 0754.57006
[4] Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. · Zbl 0731.57001
[5] F. Gantmacher, Theory of matrices, vol. 2, Chelsea, 1960. · Zbl 0088.25103
[6] A. Papadopoulos, Reseaux Ferroviares, diffeomorphismes pseudo-Anosov et Automorphismes symplectiques de l’homologie, Publ. Math. d’Orsay 83-103 (1983).
[7] Robert C. Penner, A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988), no. 1, 179 – 197. · Zbl 0706.57008
[8] R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. · Zbl 0765.57001
[9] Athanase Papadopoulos and Robert C. Penner, A characterization of pseudo-Anosov foliations, Pacific J. Math. 130 (1987), no. 2, 359 – 377. · Zbl 0602.57019
[10] W. Thurston, The geometry and topology of three-manifolds, Ann. Math. Stud., Princeton Univ. Press, Princeton, NJ (to appear).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.