# Reducible finite order mappings classes and their action on the Thurston boundary

Let $$S$$ denote the closed oriented surface of genus $$g\geq 2$$, and $$\text{Mod}(S)$$ be the mapping class group of $$S$$. Let $$f\in \text{Mod}(S)$$ be a finite order reducible element i.e. $$f$$ has a representative $$\phi\in \text{Homeo}^+(S)$$ such that $$\phi^k=$$ identity for some $$k>1$$ and $$\phi$$ preserve a multi-curve in $$S$$. There is a natural action of the cyclic group $$\text{Mod}(S)$$ on the Thurston boundary $$\mathcal{PMF}(S)$$. Is the set $$\text{Fix}(f):=\{[(\mathcal{F},\mu)]\in \mathcal{PMF}(S)|\, f.[(\mathcal{F},\mu)]=[(\mathcal{F},\mu)]\},$$ always a finite set?

No. If $$f$$ is the identity mapping class then $$\mathrm{Fix}(f)$$ is all of $$\mathcal{PMF}(S)$$.
More generally, if $$f$$ is periodic the quotient $$X = S / f$$ is an orbifold. Therefore we can take any projective measured foliation on $$X$$ and lift it back to $$S$$ to obtain a projective measured foliation on $$S$$ that is invariant under $$f$$. However, since $$f$$ is reducible, $$X$$ is not a triangle orbifold and so $$X$$ has infinitely many different projective measured foliations.