I was reading chpater (9) of the "Ergodic Theory with a view towards Number Theory" book by Manfred Einsiedler and Thomas Ward.
To be more precise, I was trying to understand the connection between the Gauss Map and the Geodesic Flow as it is illustrated in the Section 6 of the chpater (9.6 Ergodicity of the Gauss Map).
To be honest, the idea was not too much clear to me (I did not say that it is not clear at all). Therefore, I am now looking for references that illustrate the connection between the Gauss Map and the Geodesic Flow is a way that is easier than the one in this chapter.
I would be very grateful if one could suggest some good references.
See pages 317, 318 and 319.
One of the questions is:
What is the "first" visit of the geodesic flow to the set $\pi(C)$ to consider the next visit?
I do not have time to check the calculations, but the authors define a certain closed subset $C$ of the unit tangent bundle $X=UT(M)$ (of the modular orbifold). Given a point $v\in C$, the geodesic $c_v$ in $X$ defined by $v$ (so that $c_v(0)=v$) first leaves $C$ at time $t_0\ge 0$, which can be regarded as the end of the "first visit" of $C$. Formally speaking, $t_0$ is the supremum of all $t\ge 0$ such that $c_v(t)\in C$.
Then there is the smallest $t_1^- >t_0$ such that $c_v(t)\in C$. This is the "next time" $c_v$ visits $C$, more precisely, the first time $c_v$ reenters $C$.
You did not ask, but there will be $t_1^+$, the "end of the 2nd visit," where $t_1^+$ is the supremum of all $\{t \ge t_1^-: c_v(t)\in C\}$. This is the end time of the 2nd visit. And so on.