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 subcomplex $W\subset \mathcal {C}(S)$ such that $f$ fixes $W$?

Any reference is appreciate. Thanks !