  Suppose that T is a map of the Wiener space into itself, of the following
type: T=I+u where u takes its values in the Cameron-Martin space H. Assume also
that u is a finite sum of H-valued multiple Ito-Wiener integrals. In this work
we prove that if T preserves the Wiener measure, then necessarily u is in the
first Wiener chaos and the transformation corresponding to it is a rotation in
the sense of [9]. Afterwards the ergodicity and mixing of such transformations,
which are second quantizations of the unitary operators on the Cameron-Martin
space, are characterized. Finally, the ergocity of the transformation
dY_t=gamma(t)dW_t, 0 \le t \le 1 where W is n-dimensional Wiener and gamma is
non random is characterized
