# Equivalence between non-singular vector field without periodic orbits and the irrational flow on a torus.

I'm reading the book "The dynamics of vector Fields in dimension 3 - Matthias Moreno and Siddhartha Bhattacharya "

On page 6 the authors enunciate the following theorem:

Theorem:(Poincaré, Denjoy:) Every non-singular $\mathcal{C}^2$ vector field on a compact surface that has no periodic orbits is topologically equivalent to a linear flow on a torus with the irrational slope.

An interesting comment is that this result is not true for $\mathcal{C}^1$ vector fields.

I searched for this result on the Internet but I didn't find any book/paper that demonstrates the above theorem.

Does anyone know how to prove this or can give me a reference where I can learn the proof of the theorem?

N.B. I only need to know how to demonstrate the theorem when the vector field is on $\mathbb{T}^2$.

• Note that the surface must be the torus, otherwise the vector field must vanish (Lefschetz), Then see Denjoy's theorem: en.wikipedia.org/wiki/Denjoy%27s_theorem_on_rotation_number. I guess a very rough idea would be "take a circular Poincaré section, apply Denjoy's theorem, then extend the conjugation to the whole torus". – D. Thomine Jun 21 '18 at 8:29
• @D.Thomine When a vector field on $\mathbb{T}^2$ has no periodic orbits and singular points is always possible to "take a circular Poincaré Section"? Do you have any reference where can I learn such result? I'm asking this because the existence of this special section is related to another problem that I'm facing. – Matheus Manzatto Jun 22 '18 at 22:20
• See, e.g. Katok and Hasselblatt Introduction to the Modern Theory of Dynamical Systems, pp. 457-458. – user539887 Jun 23 '18 at 20:38
• @user539887 thx for this reference, you help me a lot (the section that you recommended me to read has awesome theorems!). – Matheus Manzatto Jun 24 '18 at 0:54
• I don't know ... Perhaps you could try asking a new question? – user539887 Aug 13 '18 at 5:21