of irreducible ones.

Pseudo-Anosov Elements for Connected Surfaces {#Pseudo-Anosov: Conn}
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Suppose for now that ${\Sigma}$ is a compact orientable *connected* surface such that the interior of ${\Sigma}$ admits a complete hyperbolic metric, i.e. $\chi({\Sigma})<0$. We write $g({\Sigma})$ and $b({\Sigma})$ for the genus and the number of connected components of $\partial {\Sigma}$ respectively. Thurston introduced the space $\PML({\Sigma})$ of *projective measured laminations*, equipped with a topology which makes it homeomorphic to a sphere of dimension $6g({\Sigma})+2b({\Sigma})-7$. With respect to this topology, the mapping class group $\Mod({\Sigma})$ acts naturally by homeomorphisms. $\PML({\Sigma})$ compactifies the Teichmüller space of ${\Sigma}$, coherently with respect to the actions of $\Mod({\Sigma})$.

An element $\sigma \in \Mod({\Sigma})$ is said to be *pseudo-Anosov* if the fixed-point set $\Fix(\sigma) \subset \PML({\Sigma})$ of the action of $\sigma$ on $\PML({\Sigma})$ consists of a pair of distinct measured laminations. The key property of a pseudo-Anosov element $\sigma$ is that its action on $\PML({\Sigma})$ exhibits the *north-south dynamics* with one of the fixed points as an attractor, denoted by $\scrL^+_\sigma$, and the other as a repeller, denoted by $\scrL^-_\sigma$. More precisely, for any pair of disjoint neighborhoods $U^+_\sigma$ of $\scrL_\sigma^+$ and $U^-_\sigma$ of $\scrL_\sigma^-$, we have $\sigma^{\pm N} \big( \PML({\Sigma})-U^\mp_\sigma \big) \subset U^\pm_\sigma$ 