Singular perturbations of complex polynomials and circle inversion maps

The dynamics of the family of complex functions Flz=z n+l/zd is explored. For n = d = 2 conditions are given on l ensuring that the Julia set of Fl is a Sierpinski curve. Further, a complete description of the symbolic dynamics of Fl restricted to its Julia set (in the cases where the Julia set is a Sierpinski curve) is given. For n = 2 and d = 1 the existence of a sequence of ln∈R- is shown where Fln has a superattracting n-cycle and J( Fln ) is a Sierpinski curve. The dynamics of a function derived from multiple circle inversions is also investigated. A description of the dynamics of this function restricted to its Julia set via symbolic dynamics is given. Finally, a description of the (topological and dynamcal) bifurcations that occur as this system is perturbed is presented.