Nonparametric estimation of the chaotic function and the invariant measure of a dynamical system

Let (Xt), be valued stochastic process defined by a discrete time dynamical system as Xt = [phi](Xt-1, T = 1,2,..., where [phi] is some nonlinear function preserving a probability measure say [mu], we estimate [phi] and the density -f of [mu] without using special condition on the analytical form of [phi], with nonparametric methods and some convergence rates are given.