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
  • 1
    I advise you to edit your title such that it describes the question, not the reference. – onurcanbektas Aug 8 at 13:44
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    Now posted also on MO: Question on a proof of density of periodic orbits. – Martin Sleziak Aug 9 at 15:30

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