  Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\geq 1$. Let
$\alpha_1,\alpha_2\subset S$ be two simple closed geodesics such that
$\{\alpha_1, \alpha_2\}$ fills $S$. It was shown by Thurston that most maps
obtained through Dehn twists along $\alpha_1$ and $\alpha_2$ are pseudo-Anosov.
Let $a$ be a puncture. In this paper, we study the family $\mathcal{F}(S,a)$ of
pseudo-Anosov maps on $S$ that projects to the trivial map as $a$ is filled in,
and show that there are infinitely many elements in $\mathcal{F}(S,a)$ that
cannot be obtained from Dehn twists along two filling geodesics. We further
characterize all elements in $\mathcal{F}(S,a)$ that can be constructed by two
filling geodesics. Finally, for any point $b\in S$, we obtain a family
$\mathcal{H}$ of pseudo-Anosov maps on $S\backslash \{b\}$ that is not obtained
from Thurston's construction and projects to an element $\chi\in
\mathcal{F}(S,a)$ as $b$ is filled in, some properties of elements in
$\mathcal{H}$ are also discussed.

Complex Variables;Geometric Topology