TITLE: Self-avoiding/reflecting geodesics on a convex surface
QUESTION [6 upvotes]: Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or with non-negative Gaussian curvature
at each point. Likely these various versions do not affect the question I am posing. Maybe even convexity is not required.
Let $x \in S$ be an arbitrary point on $S$, and $u$ an arbitrary direction
vector tangent to $S$ at $x$.
Shoot off a geodesic $\gamma$ from $x$ in direction $u$, and let it proceed
until it intersects its own path at some point $y$.
Rather than let the geodesic cross itself at $y$, have it instead
reflect from $\gamma$ like a mirror: angle of incidence $=$ angle of reflection.
And continue: every time $\gamma$ would cross itself, instead it reflects.
Call this $\gamma$ a reflecting geodesic.
("Self-avoiding" is eye-catching but inaccurate.)

Q. For generic $x$ and $u$, is it true that for "most" surfaces $S$,
  a reflecting geodesic $\gamma$, emanating from $x$ in direction $u$,
  converges to a point?

I would prefer not to attempt to define precisely what "most surfaces" means,
but in the polyhedral world, it would suffice for $S$ to be the convex hull
of random points in space. (Vertices would be hit with probability zero.)
I know that, without the "generic" qualifier, a reflecting geodesic
might get caught in an infinite loop.
For example, on a cube, $\gamma$ intersects itself at $90^\circ$
(essentially: because of the Gauss-Bonnet theorem):

          


   

A geodesic starting at the center
of the left-front face
eventually
intersects itself at $90^\circ$ on the bottom face.


These reflecting geodesics may seem contrived,
but something close to these came up in my research,
and I am hoping that an answer to Q might help.

REPLY [4 votes]: It is true.
If the set of limit points of your trajectory is not a single point 
it has to be a (limit set of) simple closed geodesics.
On the other hand, from Gauss--Bonnet formula, it follows that most of convex closed polyhedral surfaces do not admit simple closed geodesic.