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Limit points of lines of minima in Thurston's boundary of Teichmüller space

Raquel Diaz and Caroline Series

Algebraic & Geometric Topology 3 (2003) 207–234

arXiv: math.GT/0303108

Abstract

Given two measured laminations μ and ν in a hyperbolic surface which fill up the surface, Kerckhoff [Lines of Minima in Teichmueller space, Duke Math J. 65 (1992) 187–213] defines an associated line of minima along which convex combinations of the length functions of μ and ν are minimised. This is a line in Teichmüller space which can be thought as analogous to the geodesic in hyperbolic space determined by two points at infinity. We show that when μ is uniquely ergodic, this line converges to the projective lamination [μ], but that when μ is rational, the line converges not to [μ], but rather to the barycentre of the support of μ. Similar results on the behaviour of Teichmüller geodesics have been proved by Masur [Two boundaries of Teichmueller space, Duke Math. J. 49 (1982) 183–190].

Keywords
Teichmüller space, Thurston boundary, measured geodesic lamination, Kerckhoff line of minima
Mathematical Subject Classification 2000
Primary: 20H10
Secondary: 32G15
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Publication
Received: 17 January 2003
Accepted: 3 February 2003
Published: 26 February 2003
Authors
Raquel Diaz
Deparmento Geometría y Topología
Fac. CC. Matemáticas
Universidad Complutense
28040 Madrid
Spain
Caroline Series
Mathematics Institute
University of Warwick
Coventry CV4 7AL
UK