Dynamical invariants and parameter space structures for rational mapsReport as inadecuate


Dynamical invariants and parameter space structures for rational maps


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Abstract

For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family Fλ = z^n + λ-z^d gives a 1-parameter, n+d degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial z^n. This work presents several results related to these goals for the family Fλ, particularly regarding a structure of -necklaces- in the λ parameter plane. This structure consists of infinitely many simple closed curves which surround the origin, and which contain postcritically finite parameters of two types: superstable parameters and escape time Sierpinski parameters. First, we derive a dynamical invariant to distinguish the conjugacy classes among the superstable parameters on a given necklace, and to count the number of conjugacy classes. Second, we prove the existence of a deeper fractal system of -subnecklaces,- wherein the escape time Sierpinski parameters on the previously known necklaces are themselves surrounded by infinitely many necklaces.

Boston University Theses and Dissertations -



Author: Cuzzocreo, Daniel L. - -

Source: https://open.bu.edu/







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