A classical theorem by Denjoy asserts that every C^2 circle diffeomorphism with irrational rotation number (i.e. without periodic points) does not have a wandering interval. A consequence is that any C^2 circle diffeomorphism is topologically conjugate to a rigid rotation.
Conversely, it was shown that for every epsilon >0, there exist C^(2-epsilon) diffeomorphisms with a wandering interval. Hence, a clear break in the dynamics of circle maps occurs when viewed from the point of smoothness of these maps.
In two dimensions much less is known; simple examples show that certain dynamical phenomena that arise on the circle do not arise on for example the two-torus and vice versa. Even the topology of the wandering domains on the torus has not been systematically studied. There are examples known of diffeomorphisms of class C^(3-epsilon) with that are semiconjugate to a translation, have no periodic points and have a wandering domain with dense orbit. It is conjectured that on the torus the break in possible behaviour occurs at n=3. That is, there are no wandering domains possible for C^3 maps from the torus to itself.
There are some analogies of the nonlinearity and Schwarzian derivative cocycle in higher dimension. The project aims to investigate whether these are of use in studying problems of the kind mentioned above. |
