I am reading a note on hyperbolic surfaces (http://bicmr.pku.edu.cn/~wyang/132382/notes2.pdf). In page 35, there is a theorem stated that:
The limit set of a non-elementary Fuchsian group is the minimal G-invariant closed set in boundary of $\mathbb{D}$
. I understood the proof of this theorem well. But I don't know why from this theorem we can imply the following corollary?:
Let $G$ be a non-elementary Fuchsian group. Then the following holds:
(1) The closure of fixed points of hyperbolic elements coincides with the limit set
(2) The closure of fixed points of parabolic elements coincides with the limit set if parabolic elements exist
(3) Any orbit is dense in the limit set
Idea: In the case (1), I am trying to prove that the closure of set of repelling endpoints of hyperbolic elements in $G$, saying $\overline{\{\gamma^{+}\}}$, is a $G$-invariant.
This corollary is trivial.
(3) is immediate, as you are speaking about a minimal dynamical system (G acting on its limit set), by definition.
(1) is also easy, we just need to show that every such a fixed point, belongs to the limit set (so the whole set of fixed points is contained in the limit set, and so is its closure, and then conclude by minimality). So let $z$ be such a fixed point. There exists a geodesic defined by the hyperbolic element having $z$ as one of its ends, pick a point $x$ in this geodesic, then $g^{n}.x$ will converge to $z$, so $z$ is in the limit set.
(2) Assume that $p$ is a parabolic element, and let $z$ be a fixed point of it. The idea is that we can find hyperbolic element $g$ which normalizes $p$ (think about $p$ being unipotent and $g$ in the split Cartan subgroup) So because $p.z=z$ we have also $gpg^{-1}.z=z$ so also $g^{-1}.z$ is a fixed point for $p$, by uniqueness of fixed points, we have $g^{-1}.z=z$, so $z$ is also fixed by $g$, and then use (1).