On Representations of Certain Pseudo-anosov Maps of Riemann Surfaces with Punctures

Let S be a Riemann surface of type (p, n) with 3p + n > 4 and n ≥ 1. Let α1, α2 ⊂ S be two simple closed geodesics such that {α1, α2} fills S. It was shown by Thurston that most maps obtained through Dehn twists along α1 and α2 are pseudo-Anosov. Let a be a puncture. In this paper, we study the family 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 F(S, a) that cannot be obtained from Dehn twists along two filling geodesics. We further characterize all elements in F(S, a) that can be constructed by two filling geodesics. Finally, for any point b ∈ S, we obtain a family H of pseudo-Anosov maps on S\{b} that is not obtained from Thurston’s construction and projects to an element χ ∈ F(S, a) as b is filled in, some properties of elements in H are also discussed.