#### Algebraic and Geometric Topology 3 (2003),
paper no. 9, pages 207-234.

## Limit points of lines of minima in Thurston's boundary of Teichmueller space

### Raquel Diaz and Caroline Series

**Abstract**.
Given two measured laminations mu and nu 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 mu and nu
are minimised. This is a line in Teichmueller space which can be
thought as analogous to the geodesic in hyperbolic space determined by
two points at infinity. We show that when mu is uniquely ergodic, this
line converges to the projective lamination [mu], but that when mu is
rational, the line converges not to [mu], but rather to the barycenter
of the support of mu. Similar results on the behaviour of Teichmueller
geodesics have been proved by Masur [Two boundaries of Teichmueller
space, Duke Math. J. 49 (1982) 183-190].
**Keywords**.
Teichmueller space, Thurston boundary, measured geodesic lamination, Kerckhoff line of minima

**AMS subject classification**.
Primary: 20H10.
Secondary: 32G15.

**DOI:** 10.2140/agt.2003.3.207

**E-print:** `arXiv:math.GT/0303108`

Submitted: 17 January 2003.
Accepted: 3 February 2003.
Published: 26 February 2003.

Notes on file formats
Raquel Diaz

Deparmento Geometria y Topologia, Fac. CC. Matematicas

Universidad Complutense, 28040 Madrid, Spain

and

Caroline series

Mathematics Institute, University of Warwick

Coventry CV4 7AL, UK

Email: radiaz@eucmos.sim.ucm.es, cms@maths.warwick.ac.uk

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