Semigroups of isometries of the hyperbolic plane

Motivated by a problem on the dynamics of compositions of plane hyperbolic isometries, we prove several fundamental results on semigroups of isometries, thought of as real M\"obius transformations. We define a semigroup $S$ of M\"obius transformations to be \emph{semidiscrete} if the identity transformation is not an accumulation point of $S$. We say that $S$ is \emph{inverse free} if it does not contain the identity element. One of our main results states that if $S$ is a semigroup generated by some finite collection $\mathcal{F}$ of M\"obius transformations, then $S$ is semidiscrete and inverse free if and only if every sequence of the form $F_n=f_1\dotsb f_n$, where $f_n\in\mathcal{F}$, converges pointwise on the upper half-plane to a point on the ideal boundary, where convergence is with respect to the chordal metric on the extended complex plane. We fully classify all two-generator semidiscrete semigroups, and include a version of J{\o}rgensen's inequality for semigroups. We also prove theorems that have familiar counterparts in the theory of Fuchsian groups. For instance, we prove that every semigroup is one of four standard types: elementary, semidiscrete, dense in the M\"obius group, or composed of transformations that fix some nontrivial subinterval of the extended real line. As a consequence of this theorem, we prove that, with certain minor exceptions, a finitely-generated semigroup $S$ is semidiscrete if and only if every two-generator semigroup contained in $S$ is semidiscrete. After this we examine the relationship between the size of the `group part' of a semigroup and the intersection of its forward and backward limit sets. In particular, we prove that if $S$ is a finitely-generated nonelementary semigroup, then $S$ is a group if and only if its two limit sets are equal. We finish by applying some of our methods to address an open question of Yoccoz.