Abstract:
Topological inverse limits play an important in the theory of dynamical systems and in continuum theory. In this dissertation, we investigate classical inverse limits of Julia sets and set-valued inverse limits of arbitrary compacta. Using the theory of Hubbard trees, the trunk of the Julia set of a postcriticallly finite polynomial is introduced. Using this trunk, a characterization of indecomposability is provided for inverse limits of post-critically finite polynomials restricted to their Julia sets. Inverse limits with upper semicontinuous set-valued bonding maps are also examined. We provide necessary and sufficient conditions for inverse limits of upper semicontinuous functions to have the full projection property, answering a question posed by Ingram. The full projection property is an important tool in the study of indecomposable inverse limits. A characterization of the full projection property for arbitrary compacta is given based solely on the dynamics of the bonding functions and a second characterization is given for the class of continuum-valued maps of trees that are residual-preserving.
