$\PML({\Sigma})$ turns out to be homeomorphic to the *join* of $\PML({\Sigma}^i)$. There is a natural action of $\Mod({\Sigma})$ on $\PML({\Sigma})$, but the description of this action is rather cumbersome. For example, the action of pseudo-Anosov element $\sigma \in \Mod({\Sigma})$ still exhibits dynamical properties similar to the connected case, with an attracting neighborhood $U^+_\sigma$ and a repelling neighborhood $U^-_\sigma \subset \PML({\Sigma})$, but neither of them can arise as a neighborhood of a single projective measured lamination any more; under the homeomorphism $\PML({\Sigma}) \cong \ast_{i=1}^{c({\Sigma})} \PML({\Sigma}^i)$, certain simplexes in the join $\ast_{i=1}^{c({\Sigma})} \PML({\Sigma}^i)$ play the roles of the attractor and the repeller.

For our purposes, it is convenient to look at the action of $\Mod({\Sigma})$ on an alternative space. Following Ivanov [@Ivanov:MCGBook], we look at the natural action of $\Mod({\Sigma})$ on the space $$\PML^\sharp({\Sigma}):=\bigsqcup_{i=1}^{c({\Sigma})} \PML({\Sigma}^i)$$ instead. If an element $\sigma \in \Mod({\Sigma})$ takes each component ${\Sigma}^i \subset {\Sigma}$ to itself, it takes each component $\PML({\Sigma}^i) \subset \PML^\sharp({\Sigma})$ to itself under this action. If $\sigma$ is further assumed to be pseudo-Anosov, then its action on each component $\PML({\Sigma}^i) \subset \PML^\sharp({\Sigma})$ exhibits the honest north-south dynamics with fixed points $\scrL^\pm_{\sigma|_{{\Sigma}^i}} \in \PML({\Sigma}^i)$. Moreover, the points $\scrL^\pm_{\sigma|_{{\Sigma}^i}}$ are precisely the fixed points for the action 