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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 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 $\pi/4$

After that, they used this fact to say that: "most geodesics through $z$ are entirely contained in $U$", with figure:

I could not understand this argument. So I hope everyone will help me! I am new in this subject. This is the full proof:

   

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    $\begingroup$ Didn’t you previously ask exactly the same question? I no longer see it on the site, but deleting and reposting is (obviously) not something you should be doing. $\endgroup$ Aug 17, 2018 at 1:59
  • $\begingroup$ I apologize, just because I have been confusing with this question for a long time, I don't want to bother anyone, but I am really waiting for an answer to this question. $\endgroup$
    – user127549
    Aug 17, 2018 at 10:59
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    $\begingroup$ The idea, I think is that if $p$ is very close to the unit circle, then any geodesic through $p$ whose direction is not close to radial is very short (in Euclidean length): recall that geodesics in the unit disc are circular arcs that meet the unit circle in a right angle. $\endgroup$ Aug 17, 2018 at 16:46

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If the direction of a geodesic through $p$ is far from the radial direction, it looks something like this. The only geodesics through $p$ that are long are close to the radial direction. Getting explicit bounds is not difficult in the half plane model of hyperbolic space, but seems to be a bit more tricky in the unit disc model of the hyperbolic space.

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    $\begingroup$ Thanks, Anthony Quas, I would like to ask you about the next argument: "Since $\gamma$ preserves angles and the same observation applies to geodesics through $\gamma (z)$ we can, in fact, find a geodesic k through $z$ and cointained in $U$ such that $\gamma k$ is contained in $V$". Since $\gamma c$ is different to $c$, so how the angle-preserving can be applied in this case? There is also a similar technicality used by Lee Mosher in his answer in math.stackexchange.com/questions/1892970/…, but I don't know how to find K, C exactly. $\endgroup$
    – user127549
    Aug 20, 2018 at 11:47
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    $\begingroup$ The idea is that $\gamma$ maps geodesic through $z$ to geodesic through $\gamma(z)$. These are parameterized by the angle, but angle is preserved. Since 3/4 of geodesic through $z$ are short and 3/4 of geodesic through $\gamma(z)$ are short, there exist geodesics $\kappa$ through $z$ which are short and such that $\gamma\kappa$ is short also. $\endgroup$ Aug 21, 2018 at 19:23

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