respectively for all sufficiently large $N$.

We remark that pseudo-Anosov elements are always aperiodic. Furthermore, as mentioned earlier, one of the main results in Thurston’s work [@Thurston:Surfaces] is that irreducible elements in $\Mod({\Sigma})$ are precisely pseudo-Anosov ones; see [@FLP:Thurston] for the detail.

Pseudo-Anosov Elements for Disonnected Surfaces {#Pseudo-Anosov: Disconn}
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Let us now allow a compact orientable surface ${\Sigma}=\bigsqcup_{i=1}^{c({\Sigma})}{\Sigma}^i$ to be *disconnected*, where $c({\Sigma})$ denotes the number of connected components of ${\Sigma}$ and ${\Sigma}^i$ denotes each connected component. As in the case of connected surfaces, we will assume that the interior of ${\Sigma}$ admits a complete hyperbolic metric, i.e. $\chi({\Sigma}^i)<0$ for each ${\Sigma}^i$.

Every element $\sigma \in \Mod({\Sigma})$ has a nontrivial power that takes each connected component ${\Sigma}^i$ to itself. An element of $\Mod({\Sigma})$ is said to be *pseudo-Anosov* if the restriction of such a power to each component ${\Sigma}^i$ is a pseudo-Anosov element in $\Mod({\Sigma}^i)$. It is easy to see that this notion of pseudo-Anosov elements is well-defined, and they are necessarily aperiodic. Furthermore, one can check that irreducible elements of $\Mod({\Sigma})$ are precisely pseudo-Anosov ones. With some care, the space $\PML({\Sigma})$ of projective measure laminations can be defined. As shown by Ivanov [@Ivanov:Tits-Margulis-Soifer] and McCarthy [@McCarthy:Tits], the space 