  Let $X$ be an infinite hyperbolic surface endowed with an upper bounded
geodesic pants decomposition. Alessandrini, Liu, Papadopoulos, Su and Sun
\cite{ALPSS}, \cite{ALPS} parametrized the quasiconformal Teichm\"uller space
$T_{qc}(X)$ and the length spectrum Teichm\"uller space $T_{ls}(X)$ using the
Fenchel-Nielsen coordinates. A quasiconformal map $f:X\to Y$ is said to be {\it
asymptotically conformal} if its Beltrami coefficient $\mu
=\bar{\partial}f/\partial f$ converges to zero at infinity. The space of all
asymptotically conformal maps up to homotopy and post-composition by conformal
maps is called "little" Teichm\"uller space $T_0(X)$. We find a parametrization
of $T_0(X)$ using the Fenchel-Nielsen coordinates and a parametrization of the
closure $\overline{T_0(X)}$ of $T_0(X)$ in the length spectrum metric. We also
prove that the quotients $AT(X)=T_{qc}(X)/T_0(X)$,
$T_{ls}(X)/\overline{T_{qc}(X)}$ and $T_{ls}(X)/\overline{T_0(X)}$ are
contractible in the Teichm\"uller metric and the length spectrum metric,
respectively. Finally, we show that the Wolpert's lemma on the lengths of
simple closed geodesics under quasiconformal maps is not sharp.
