Complex torus, KAM theorem and diffeomorphisms

Question

I am sorry if this is obvious, but I needed clarify few things about complex torus.

In two dimensions the group of large diffeomorphisms on  complex torus are defined by the modular group, whose elements are represented by the elements of  \(SL(2, \mathbb{Z})\). I understand that  given two complex tori defined by the complex vectors:\((\omega_1, \, \omega_2 )\) and \((\omega'_1,\, \omega'_2)\),we can infer  that they are equivalent if the vectors can be obtained from one other via modular transformation. 

1. Suppose we have a non-contractable loop on the torus with the winding numbers \(m\) and \(n\). What exactly happens to them under a modular transformation? Do they change the same way as \((\omega_1, \, \omega_2 )\)?

2. One of the results of KAM theorem is that quasiperiodic trajectories on the real plane share the same topology with a torus. Can we make a similar statement if the trajectories live in the complex plane i.e do complex quasiperiodic orbits have the topology of a complex torus? 

Comments

Can you please clarify your question? When considering periods on the complex line it is regarded as a real 2-space, so the complex structure seems irrelevant to me.

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@Arnold Neumaier are you referring to the second question  by complex line ?  

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The complex line is the simplest example of a complex manifold; so I used it for illustration. In general, there is a KAM theorem for the real form of the manifold and associated KAM tori. What do you ask that is not already in the real KAM theory on the real form of the manifold?

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@Arnold Neumaier I was wondering whether one can define an equivalence relation between complex orbits with different winding numbers. If the complex orbits are topologically equivalent to complex torus then modular transformations, I think, might be a natural way to define such equivalence. Even if this makes sense,  I am not sure this can be done on the real torus.