| In this dissertation we discuss topological tools for studying the Julia sets of rational functions. The first results are concerned with pathological cases. To that end, we characterize indecomposable subcontinua of the plane in a way which could prove useful for recognizing indecomposable rational Julia sets, especially those with buried points. We then prove that any counter-example of Makienko's conjecture has an indecomposable continuum as its Julia set, and relatedly study the structure of Julia sets which are irreducible continua. In the direction of understanding less pathological examples, we prove for all connected polynomial Julia sets that there exists a finest monotone semiconjugacy to a dynamical system on a locally connected phase space. Finally, we construct a (realizable) locally connected model for a cubic polynomial with a wandering branch point.;Keywords. Complex dynamics, continuum theory, indecomposable continuum. |
