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 $\mathbb{T}^2$.