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Limit points of lines of minima in Thurston’s boundary of Teichmüller space. (English) Zbl 1066.32020
Let $$S$$ be a hyperbolic surface and let $$\mu$$ and $$\nu$$ be two measured geodesic laminations that fill up $$S$$. We denote by $$l_{\mu}$$ and $$l_{\nu}$$ the geodesic length functions defined by $$\mu$$ and $$\nu$$ on the Teichmüller space $$\mathcal T$$ of $$S$$. S. P. Kerckhoff observed in [Duke Math. J. 65, No. 2, 187–213 (1992; Zbl 0771.30043)] that for any $$s\in(0,1)$$ the function $$(1-s)l_{\mu}+ sl_{\nu}$$ has a unique global minimum $$m_s$$ on $$\mathcal T$$, and he defined the line of minima $${\mathcal L}_{\mu,\nu}$$ to be the family of these minima. He stated the following fact: choosing the support of $$\mu$$ to be a union of at least two disjoint closed geodesics, one can obtain an example of a line of minima $${\mathcal L}_{\mu,\nu}$$ which does not converge to the projective equivalence class of $$\mu$$ in Thurston’s compactification $${\mathcal PMF}$$ of $$\mathcal T$$.
In this paper, the authors give a proof of that statement. They show taking line of minima $${\mathcal L}_{\mu,\nu}$$ with $$\mu=\sum_ {i=1}^n a_i\alpha_i$$, where $$\alpha_i$$ is a simple closed geodesic and $$a_i>0$$ for all $$i=1,\ldots, n$$, then the point $$m_s$$ converges as $$s\to 0$$ to the projective class of the lamination $$\sum \alpha_i$$, and not to that of $$\sum a_i\alpha_i$$. They also prove that if $$\mu$$ is uniquely ergodic, then $$\lim_{s\to 0} m_s=[\mu]\in{\mathcal PMF}$$. The results parallel results of Howard Masur, obtained in his paper [Duke Math. J. 49, 183–190 (1982; Zbl 0508.30039)], in which he proves a similar behaviour for Teichmüller geodesics.

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30F60 Teichmüller theory for Riemann surfaces
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