Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is positive. Consider a pants decomposition containing $x$. Consider the Fenchel-Nielsen coordinate of the Teichmuller space with respect to this pants decomposition. Let $l\_{\chi\_s}(y)$ denote the length function of $y$ with respect to the left hand side twist parameter $s$, associated to the the curve $x$. It seems that there exist a $s\_0$ such that $l\_{\chi\_s}(b)$ is an increasing function for $s>s\_0$. Is this true? If yes, what is the proof?


Thanks in advance.


PS 1: I know that the function $l\_{\chi\_s}(b)$ is strictly convex.



Is the length function associated with the twist parameter an increasing function?
Yes, this is shown in Kerckhoff's paper: 
The Nielsen Realization Problem
Steven P. Kerckhoff
Annals of Mathematics
Second Series, Vol. 117, No. 2 (Mar., 1983), pp. 235-265


Proposition 3.5


