Some topological properties of Julia sets of rational iterations

In this thesis we study the topological properties of the Julia sets generated by iterating a single rational function or a couple of rational functions randomly. Chapter 1 provides some background information and Chapter 2 is a preparation for the rest of this thesis. In chapter 3, the dynamics of rational semigroup is considered. We prove that if the finite postcritical set is bounded then the Julia set of the finitely generated polynomial semigroup is connected. An example is given to show that the converse is not true. In Chapter 4, we study the buried points problem by relating it to indecomposable continuum. By improving a theorem of Beardon in 1991, we show that if a Julia set of a rational function has buried component, then there must be uncountably many buried components. If we assume that the rational function concerned there has exactly two critical points and is of degree ≥ 3, for this class of rational functions, we are able to extend Qiao's work in 1997. Specifically we prove that if a rational function of this type has no completely invariant domain under the second iterate and its Julia set contains no buried points, then the Julia set is a Lakes of Wada continuum, which is either indecomposable or the union of two indecomposable continua