A parabolic representation of the free group is one in which the images of both generators are parabolic elements of $PSL(2,\IC)$. The Riley slice is a closed subset ${\cal R}\subset \IC$ which is a model for the moduli space of parabolic, discrete and faithful representations. The complement of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at the boundary, corresponding to parabolic discrete and faithful representations of rigid subgroups of $PSL(2,\IC)$. Recent work of Aimi, Akiyoshi, Lee, Oshika, Parker, Lee, Sakai, Sakuma \& Yoshida, have topologically identified all these groups. Here we give the first substantive properties of the nondiscrete representations using ergodic properties of the action of a polynomial semigroup and identifying the Riley slice as the ``Julia set’’ of this dynamical system. We prove a supergroup density theorem: given any irreducible parabolic representation of $F_2$ whatsoever, {\em any} non-discrete parabolic representation has an arbitrarily small perturbation which contains that group as a conjugate. Using these ideas we then show that there are nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes of parabolic generators.
