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Limit points of lines of minima in Thurston's boundary of
Teichmüller space
Raquel Diaz and Caroline Series
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Algebraic & Geometric Topology 3
(2003) 207–234
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Abstract
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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].
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Keywords
Teichmüller space, Thurston
boundary, measured geodesic lamination, Kerckhoff line of
minima
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Mathematical Subject Classification
Primary: 20H10
Secondary: 32G15
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Publication
Received: 17 January 2003
Accepted: 3 February 2003
Published: 26 February 2003
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