Cycles in Newton's means

In the dynamical system defined by Newton's means for n complex variables, n ≥ 2 there are invariant, planar curves with (chaotic) dynamics conjugated to the dynamics of z → zn on the unit circle in the complex plane. The are not many explicit examples of multidimensional, noninvertible dynamical systems with interesting dynamics which can be understood with rather elementary tools. The beauty of symmetric polynomials and their connections to many fields of mathematics make them worth considering also as dymanical systems. To our astonishment, although the behaviour of the iterations of symmetric means for positive initial points was known since long, the cyclic features were not and turned out to be surprising not only for analysts but for algebraists as well. © Springer-Verlag 2001.