Regularization of Brouwer translation theorem.

The celebrated Brouwer translation theorem asserts that for a preserving orientation fixed point free homeomorphism of the plane, each point belongs to an invariant region where the dynamics is continuously conjugate to a translation. In this work we prove that if we start with a C m, m ∈ N ∪ {∞}, diffeomorphism then the referred conjugacy has the same kind of regularity