# Research projects

## Project CEMAPRE internal

 Title Almost linearization of irrational toroidal flows Participants José Pedro Gaivão, João Lopes Dias (Principal Investigator) Summary Irrational flows on the torus are generated by vector fields with a unique rotation vector that is rationally independent. In dimension two, by taking the return map to a section of the flow, they correspond to circle diffeomorphisms with irrational rotation number. Under conditions, they are linearizable in the sense that in some coordinates the flow is a straight line with irrational direction. So, they are minimal, i.e. all orbits are dense in the torus. More general notions are notoriously important for applications, namely the almost linearization. In this case, one can turn the flow arbitrarily close to a straight line. It is expected that this holds for much more general conditions.