# On a proof of density of periodic orbits of the geodesic flow on the unit tangent bundle of closed surface

In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following:

Theorem: Let $\Gamma$ be a discrete group of fixed-point-free isometries of $\mathbb{D}$ such that $M:=\Gamma \setminus \mathbb{D}$ is compact. Then the periodic orbits of the geodesic flow on $SM$ are dense in $SM$.

In the proof, they stated without proof a fact that:

given $\epsilon >0$ there exists $\delta>0$ such that when $p \in \mathbb{D}$ is in a $\delta$-neighborhood of $\partial \mathbb{D}$ then any two geodesics through $p$ of Euclidean length greater than $\epsilon$ have a mutual angle of at most $\pi/4$

My first question is how to prove this fact?

After that, they used this fact to say that: "most geodesics through $z$ are entirely contained in $U$". I could not understand this argument. So I hope everyone will help me! I am new in this subject. This is the full proof enter image description here

Postscript: About my first question, I proved this fact by considering the worst case.

• Clearly the fact stated without proof has to do with properties of $\mathbb{D}$, which is the context you are talking about denotes the Poincare disk. – uniquesolution Aug 8 at 9:34
• yes, this fact is only a property of $\mathbb{D}$ but for me I don't know how to prove it – Student Aug 8 at 9:36
• I advise you to edit your title such that it describes the question, not the reference. – onurcanbektas Aug 8 at 13:44
• Now posted also on MO: Question on a proof of density of periodic orbits. – Martin Sleziak Aug 9 at 15:30