  We give a new very concrete description of the C*-envelope of the tensor
algebra associated to multivariable dynamical system. In the surjective case,
this C*-envelope is described as a crossed product by an endomorphism, and as a
groupoid C*-algebra. In the non-surjective case, it is a full corner of a such
an algebra. We also show that when the space is compact, then the C*-envelope
is simple if and only if the system is minimal.
