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Summary: Diophantine and ergodic foliations on
surfaces
Curtis T. McMullen
6 December 2011
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Diophantine and recurrent laminations . . . . . . . . . . . . . 3
3 Ergodicity of pseudo-Anosov foliations . . . . . . . . . . . . . 9
4 Ergodic components and stable curves . . . . . . . . . . . . . 11
A Appendix: Orbits and measures . . . . . . . . . . . . . . . . . 13
1 Introduction
This paper gives a topological characterization of Diophantine and recurrent
laminations on surfaces. It also establishes an upper bound for the number
of ergodic components of a measured foliation. Taken together, these results
give a new approach to Masur's theorem on unique ergodicity.
Teichm¨uller rays. Let Mg,n denote the moduli space of Riemann surfaces
of genus g with n punctures. Consider the Teichm¨uller ray
: [0, ) Mg,n
generated by a holomorphic quadratic differential q = q(z) dz2 on X Mg,n.
If there exists a compact set K Mg,n such that (t) K for all t, we say
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