  Let $\mathrm{Mod}(S_g)$ denote the mapping class group of the closed
orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of
$\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ denote the set of fixed points
induced by the action of $H$ on the Teichm\"{u}ller space
$\mathrm{Teich}(S_g)$. When $H$ is cyclic with $|H| \geq 3$, we show that
$\mathrm{Fix}(H)$ admits a decomposition as a product of two-dimensional strips
at least one of which is of bounded width. For an arbitrary $H$ with at least
one generator of order $\geq 3$, we derive a computable optimal upper bound for
the restriction $\mathrm{sys} : \mathrm{Fix}(H) \to \mathbb{R}^+$ of the
systole function. Furthermore, we show that in such a case, $\mathrm{Fix}(H)$
is not symplectomorphic to the Euclidean space of the same dimension. Finally,
we apply our theory to recover three well-known results, namely: (a) Harvey's
result giving the dimension of $\mathrm{Fix}(H)$, (b) Gilman's result that $H$
is irreducible if and only if the corresponding orbifold is a sphere with three
cone points, and (c) the Nielsen realization theorem for cyclic groups.
