I'm reading the book "The dynamics of vector fields in dimension 3" - "Matthias Moreno and Siddhartha Bhattacharya". On page 10 the authors define
Hopf flow on $\mathbb{S}^3$: We identify $\mathbb{S}^3\subset \mathbb{C}^2$ with $\{ (z_1,z_2):|z_1|^2 + |z_2|^2 = 1\}$, and we define the flow $\phi$ on $\mathbb{S}^3$ by $$\phi^t(z_1,z_2)= (e^{2i\pi t} z_1 , e^{2i\pi t} z_2). $$
The author doesn't write this in the book, but I think that the Hopf vector field is $H: \mathbb{S}^3 \to T \mathbb{S}^3$; $H(a+ib,c + id) = ( (a+ib, c+ id) , 2\pi(-b + i a ,-d + ic)$.
After this, the book does the following procedure (sorry for the big image).
My problem is in the demonstration of the Lemma 1, I really don't understand what he did.
My Doubts about the Lemma's demonstration: 1) I think in the underlined part in blue is $\pi^{-1} (C)$ instead of $C$ (because doesn't make sense an orbit of $X_i$ lies in $C$).
2) The underlined red part doesn't make much sense to me, how these $h$'s are taken? Which periodic orbits were used to construct those Poincare's first return map?
3) And finally, about the green underlined text, anyone knows where can I find a demonstration of this property?
Can anyone help me to understand this demonstration or propose another reference to I see the demonstration of this theorem (Seifert's Theorem)?