/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

This article was widely circulated as a preprint, about 12 years ago. At that time the Bulletin did not accept research announcements, and after a couple of attempts to publish it, I gave up, and the preprint did not find a home. I very soon saw that there were many ramifications of this theory, and I talked extensively about it in a number of places. One year I devoted my graduate course to this theory, and notes of Bill Floyd and Michael Handel from that course were circulated for a while. The participants in a seminar at Orsay in 1976-1977 went over this material, and wrote a volume [FLP] including some original material as well. Another good general reference, from a somewhat different point of view, is a set of notes of lectures by A. Casson, taken by S. Bleiler [CasBlei]. There are by now several alternative ways to develop the classification of diffeomorphisms of surfaces described here. At the time I originally discovered the classification of diffeomorphism of surfaces, I was unfamiliar with two bodies of mathematics which were quite relevant: first, Riemann surfaces, quasiconformal maps and Teichmiiller's theory; and second, Nielsen's theory of the dynamical behavior of surface at infinity, and his near-understanding of geodesic laminations. After hearing about the classification of surface automorphisms from the point of view of the space of measured foliations, Lipman Bers [Bersl] developed a proof of the classification of surface automorphisms from the point of view of Teichmüller theory, generalizing Teichmiiller's theorem by allowing the Riemann surface to vary as well as the map. Dennis Sullivan first told me of some neglected articles by Nielsen which might be relevant. This point of view has been discussed by R. Miller, J. Gilman, M. Handel and me. The analogous theory, of measured laminations and 2-dimensional train tracks in three dimensions, has been considerable development. This has been applied to reinterpret some of Haken's work, to classify incompressible surfaces in particular classes of 3-manifolds in papers by me, Hatcher, Floyd, Oertel and others in various combinations. Shalen, Morgan, Culler and others have developed the related theory of groups acting on trees, and its relation to measured laminations, to define and analyze compactifications of representation spaces of groups in SL(2, C) and SO(n, 1); this has many interesting applications, including the theory of incompressible surfaces in 3-manifolds.

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On the geometry and dynamics of diffeomorphisms of surfaces

Interval Exchange Transformations and Measured Foliations

The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. 
In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov. In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface. 
We also show that the action of pseudo-Anosov mapping classes on the complex is hyperbolic, with a uniform bound on translation distance.

Geometry of the complex of curves I: Hyperbolicity

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian representations. The authors' previous result that the complex is delta-hyperbolic was hard to apply because the complex is not locally finite; in this paper some tools are developed for overcoming this problem, and a combinatorial mechanism introduced which describes sequences of elementary moves in the graph of markings on a surface. These tools are applied to give a family of quasi-geodesic words in the Mapping Class Group, and a linear bound on the shortest word conjugating two conjugate pseudo-Anosov elements. 
A basic tool in the analysis is a family of subsurface projections, which are roughly analogous to closest-point projections to horoballs in classical hyperbolic space. These projections have a strong contraction property which makes it possible to tie together the geometry of the complex and that of the (infinite) subcomplexes that arise as links of vertices. The resulting layered structure of the complex is controlled by means of a combinatorial device called a hierarchy of geodesics, which is the central construction of the paper.

Geometry of the complex of curves ii: hierarchical structure

Geometry of the complex of curves II: Hierarchical structure

This paper concerns the interplay between the complex structure of a Riemann surface and the essentially Euclidean geometry induced by a quadratic differential. One aspect of this geometry is the " trajectory structure" of a quadratic differential which has long played a central role in Teichmfiller theory starting with Teichmiiller's proof of the existence and uniqueness of extremal maps. Ahlfors and Bers later gave proofs of that result. In other contexts, Jenkins and Strebel have studied quadratic differentials with closed trajectories. Starting from the dynamical problem of studying diffeomorphisms on a C ~ surface M, Thurston [17] invented measured ]ol~t io~. These are foliations with certain kinds of singularities and an invariantly defined transverse measure. A precise definition is given in Chapter I, w 1. This notion turns out to be the correct abstraction of the trajectory structure and metric induced by a quadratic differential. In this language our main statement says that given any measured ]oliation F on M and any complex structure X on M, there is a unique quadratic diHerential on the Riemann surface X whose horizontal trajectory structure realizes F. In particular any trajectory structure on one Riemann surface occurs uniquely on every Riemann surface of that genus. In the special case when the foliation has closed leaves, an analogous theorem was proved by Strebel [15]. Earlier Jenkins [13] had proved that quadratic differentials with closed trajectories existed as solutions of certain extremal problems. We deduce Strebel's theorem from ours in Chapter I, w 3. By identifying the space of measured foliations with the quadratic forms on a fixed Riemann surface, we are able to give an analytic and entirely different proof of a result of Thurston's [17]; that the space of projective classes of measured foliations is homeomorphic to a sphere. This is also done in Chapter I, w 3.

/pdf/quadratic-differentials-and-foliations-1rw86vv3fv.pdf

Howard Masur

Papers

Interval Exchange Transformations and Measured Foliations

Geometry of the complex of curves I: Hyperbolicity

Geometry of the complex of curves ii: hierarchical structure

Geometry of the complex of curves II: Hierarchical structure

Quadratic differentials and foliations