Hi, everyone. 

Assume $S$ is a genus at least 2 orientable closed surface. And there is a simplical complex 
defined on $S$ called Curve complex. 

It is well known that any automorphism of surface $S$ acts on curve complex $\mathcal {C}(S)$ isometrically.

Now I want to know that

For any periodic and irreducible automorphism $f: S\rightarrow S$,  is there a connected subcomplex $W\subset \mathcal {C}(S)$ with infinite diameter such that $f(W)=W$?


Staylor constructed such a $W$. Now there is a handlebody $H$ such that $\partial H=S$.
Let $f$ be as above, $W$ be the disk complex of $H$.

Now I wonder: 

Is it still possible that $f(W)=W$?