Cycles in Newton's means

In the dynamical system defined by Newton's means for n complex variables, n greater than or equal to 2 there are invariant, planar curves with (chaot ic) dynamics conjugated to the dynamics of z --> z(n) on the unit circle in the complex plane. There are not many explicit examples of multidimensional, noninvertible dyn amical systems with interesting dynamics which can be understood with rathe r elementary tools. The beauty of symmetric polynomials and their connectio ns to many fields of mathematics make them worth considering also as dynami cal systems. To our astonishment, although the behaviour of the iterations of symmetric means for positive initial points was known since long, the cy clic features were not and turned out to be surprising not only for analyst s but for algebraists as well.