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?

Answer 1

For the first question, the answer is yes. Consider the torus as $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$, and a loop with winding numbers $(n,m)$ as the image in this quotient of the line joining $0$ and $n+\tau m$ in $\mathbb{C}$. You can then see $SL(2,\mathbb{Z})$ as the matrix of an endomorphism of $\mathbb{C}$ viewed as an $\mathbb{R}$-vector space with basis $(1,\tau)$. Obviously this matrix transforms in the same way the point $n+\tau m$ and the periods of the torus.

Answer 2

I think you are mixing up a few concepts, although with some care, sense can be made of most of it.

A large diffeomorphism of a given differential manifold is an isotopy class of diffeomorphisms of that manifold. This means that two diffeomorphisms correspond to the same large diffeomorphism if they can be transformed into one another in a differentiable way.

As an example (and in the case of the torus the example) of unequal large diffeomorphisms, consider different Dehn twists.

When you go to complex tori, you are talking about much more structure than a differentiable structure. In determining their equivalence, the modular group does play a role, but that doesn't have anything to do with the large diffeomorphism group (at most in a very indirect way). Your two complex tori are equivalent when the lattices generated by $(\omega_1, \, \omega_2)$ and by $(\omega'_1, \, \omega'_2)$ differ by a complex number. The lattices are equal exactly when one can be transformed into the other by a basis transform, which corresponds to an element of $GL(2,\Bbb Z)$ (note that we are talking about linear combinations with integer coefficients). These have determinant $\pm 1$, so if we restrict to ordered bases we get $SL(2,\Bbb Z)$, the modular group. Note that in that case we don't merely have equivalence, we have equality.

Note that the lattices don't have to be equal for biholomorphic equivalence, only homothetic.

To go to your first question: the group of large diffeomorphisms doesn't transform loops into loops (only homotopy classes of loops). It does however act on the complex plane, so if your torus is defined by the lattice $(\omega_1, \omega_2)$, a modular transformation determines a diffeomorphism mapping loops into loops, acting on homotopy classes like user40085 describes, and indeed this corresponds to the action by large diffeomorphisms on homotopy classes of loops.

As for the second question: I am by no means an expert on KAM theory, but I think you'll have to be more precise. A trajectory is a curve, never a torus, even in phase space. In fact, I think that the theorem states that such trajectories lie on a torus in phase space. Note that all tori, complex or not, are topologically equivalent, so your question as it stands is not interesting. Obviously you can give any torus (and any plane) a complex structure, so that is not what you are interested in either. If it is really complex sturctures that you're interested in, what could be an interesting question is if such a structure would or could be invariant under canonical transformations (I would think not, but I don't know). You could also ask if there is a natural complex structure on the tori on which such trajectories fall, e.g. if you are in the complex plane, phase space can be identified with $\Bbb C^2$. Are the tori on which quasiperiodic trajectories fall complex submanifolds of it?