Some questions on partial pseudo anosov maps

Question

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：

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

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

3. 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!

Answer 1

In answering all of your questions I am going to assume that $S$ has a hyperbolic structure, and that the stable and unstable laminations are geodesic laminations. This point of view is explained in the book of Casson and Bleiler. The answers given here can be derived from what one learns in that book.

Question 1: No it does not spiral. In a hyperbolic structure on $S$, on either side of $C$ there is a subsurface of $S$ that looks like a crown with the points of the crown removed, with $C$ being the circle component of the boundary of the crown, with the top edges of the crown being leaves of the stable lamination, and with the interior of the crown disjoint from the stable lamination. The unstable lamination has a similar appearance.

Question 2: In answering this question (and the next) I am going to assume that by ``$f^n(\alpha)$'' you really mean the simple closed geodesic which is isotopic to $f^n(\alpha)$.

$f^n(a) \cap S_C$ is a union of long subsegments that stay close to the stable lamination in the $C^1$-topology, together with a finite number (independent of $n$) of subsegments that stay in the complement of the stable lamination.

Question 3. You should make a stronger hypothesis: not only should $C \cap \partial F$ be nonempty, but their intersection should be ``essential'', meaning that no component of $S - (C \cap \partial F)$ is a bigon. With that assumption, yes, this is always true, regardless of whether $C$ is separating. The reason is that you can assume, by isotoping $C$, that $\partial C$ is geodesic, and it follows that $\partial C$ intersects the stable lamination transversely; now apply Question 2 and the fact that leaves of the stable lamination are dense.