of $\sigma$ on $\PML^\sharp({\Sigma})$. For any pair of disjoint neighborhoods $U^+_\sigma$ of $\big\{\scrL^+_{\sigma|_{{\Sigma}^i}} \big\}_{i=1}^{c({\Sigma})}$ and $U^-_\sigma$ of $\big\{\scrL^-_{\sigma|_{{\Sigma}^i}} \big\}_{i=1}^{c({\Sigma})}$, we have $\sigma^N \big( \PML^\sharp({\Sigma})-U^-_\sigma \big) \subset U^+_\sigma$ and $\sigma^{-N} \big( \PML^\sharp({\Sigma})-U^+_\sigma \big) \subset U^-_\sigma$ for all sufficiently large $N$.

Subgroups of Mapping Class Groups {#Subgroups}
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Ivanov further generalized Thurston’s theory to subgroups of $\Mod({\Sigma})$ in order to study their structures [@Ivanov:MCGBook], and obtained a classification of subgroups that parallels the classification of elements of $\Mod({\Sigma})$.

-   An isotopy class $\calA$ of an essential (possibly empty) multi-loop is said to be a *reduction system* for $\Gamma$ if it is a reduction system for every element $\sigma \in \Gamma$, i.e. if every element $\sigma \in \Gamma$ fixes $\calA$.

-   A subgroup $\Gamma$ is said to be *reducible* if it admits a *non-empty* reduction system $\calA$; it is said to be *irreducible* otherwise.

By definition, an element $\sigma \in \Mod({\Sigma})$ is reducible (resp. irreducible) if and only if the cyclic subgroup $\langle \sigma \rangle < \Mod({\Sigma})$ is reducible (resp. irreducible). More generally, the above definitions suggest that (i) the analogue of periodic elements are finite subgroups, (ii) the analogue of aperiodic reducible elements are infinite reducible subgroups, and (iii) the analogue of 