Density of orbits of action on the boundary of a convex set

Question

Let $X$ be a compact convex subset of the plane. Let $c_1$, $c_2$ and $c_3$ be non-collinear points in the interior of $X$.

For every point $x$ on the boundary $\partial X$, you can draw the line from $x$ through $c_1$ and continue it until it meets $\partial X$ in another point, which I'll call $f_1(x)$. This gives a homeomorphism $f_1: \partial X \to \partial X$. Define $f_2, f_3: \partial X \to \partial X$ similarly. For example:

Let $G$ be the subgroup of the group of self-homeomorphisms of $\partial X$ generated by $f_1$, $f_2$ and $f_3$. Then $G$ acts in a natural way on $\partial X$.

Question Is the $G$-orbit of every point of $\partial X$ dense in $\partial X$?

Equivalently, is it true that every closed subset $A$ of $\partial X$ that's completely invariant under each $f_i$ (meaning that $A = f_1 A = f_2 A = f_3 A$) is either empty or the whole of $\partial X$?

Answer 1

Consider the case when $X$ is a closed round disk in the plane with the interior (open disk) $H$ and boundary circle $C$. I will regard $H$ as a Klein model on the hyperbolic plane, whose isometries are projective transformations preserving $H$. Then, given a point $c\in H$ and defining the corresponding involution $\theta_{c}: C\to C$ as in your question, one checks that $\theta_{c}$ is the restriction of a projective involution of $X$, a hyperbolic isometry of $H$. (I will leave it to you to verify this.) Let $\Gamma$ be a subgroup of the isometry group of $H$ generated by three involutions $\theta_{c_i}, i=1,2,3$. For every point $x\in C$, the accumulation set of the orbit $\Gamma x$ is the limit set $\Lambda$ of $\Gamma$. It turns out that $\Lambda$ does not depend on $x$ and if $\Gamma$ is discrete, then $\Lambda$ equals $C$ if and only if $H/\Gamma$ has finite area ($\Gamma$ is a lattice). If $\Gamma$ is discrete then $\Lambda$ either equals $C$, or is homeomorphic to the Cantor set or consists of at most two points.

Let $H_i$ be hyperbolic half-planes in $H$ bounded by lines $L_i$ in $H$ through the points $c_i$. If one can find $H_i$'s which are pairwise disjoint, then the complement $F$ to their union is a fundamental domain of $\Gamma$ acting on $H$ (in particular, $\Gamma$ is discrete). The hyperbolic area of $F$ is the area of $H/\Gamma$. If $L_i$'s are not just disjoint but (at least) two of them have disjoint closures in $X$, then $F$ has infinite area. Thus, to find a counter-example to your conjecture, it suffices to take $c_i$'s which are sufficiently close to three pairwise distinct points in $C$.

In my answer I used standard results on Fuchsian groups. You can find detailed proofs for instance in Beardon's book "Geometry of discrete groups." (Apart from the discussion of the Klein model.)