Hipoelipticidade global de campos vetoriais no toro TN

Abstract

In this work, we will see that if the transpose operator of a smooth real vector field L defined on the N-dimensional torus, regarded as a linear differential operator with coefficients in C1(TN), is globally hypoelliptic, then there exists a vector field with constant coefficients L0 such that L and L0 are C1-conjugated, with such constants satisfying a condition called Diofantina (*). We will also show the converse of this fact, that is, if there is a coordinate system such that in this new system L has constant coefficients with such constant satisfying the Diophantine condition (*) then its transpose L* is globally hypoelliptic. We will see that the Diophantine condition implies that the flow generated by the field, regarded as a Dynamical system is minimal.