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

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

References

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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

Volume 3, issue 1 (2003)

Raquel Diaz and Caroline Series