Geometry of julia sets of complex polynomail z[superscript n]+c

Abstract

This thesis is intended to estimate the upper bounded of such that Julia sets of complex polynomials of the form z[superscript n]+c are simple closed curves when n = 2, 3, 4, ... Moreover, we study the geometric properties of these Julia sets. We know that Julia sets of complex polynomials of the form z[superscript n]+c are simple closed curves provided <1/4. We expect the same phenomenon, i.e. Julia sets of complex polynomials of the form z[superscript n]+c are simple closed curves if is small enough. However, they are far from being smooth; indeed, they contain no smooth arcs at all