Let $S$ be an orientable closed 2-surface with genus at least 2 and $C$ be a non-separating essential simple closed curve in $S$. Denote $S_{C}=S-N(C)$. Let $f$ be a pseudo anosov map of $S_{C}$. Hence $f$ induces a natural homeomorphism of $S$ which fix the $C$ pointwise, still denoted by $f$.

My question are：

What does the stable lamination of $f$ look like? Does it spiral around the $C$ in $S$?

For any simple closed curve $\alpha\cap C\neq \emptyset$, what does $f^{n}(\alpha)\cap S_{C} $ look like when $n\rightarrow +\infty$?

- Suppose there is an essential proper subsurface $F\subset S$, i.e., $\partial F\neq \emptyset$ are essential in $S$ and $C\cap \partial F\neq \emptyset$. For any component $C\cap F$ , is there a component of $f^{n}(\alpha)\cap F$ disjoint from it when $n$ is large enough? When $C$ is separating, is it still true?

Thanks!