Date of Award
2026
Degree Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics and Applied Mathematical Sciences
First Advisor
Araceli Bonifant
Abstract
We continue the study of the periodic curves introduced by John Milnor in order to facilitate the study of the cubic polynomial parameter space. The boundary of these slices exhibit many connections with the quadratic polynomial parameter space, and we investigate the ways in which results from the study of the Mandelbrot set can be upgraded to the study of the connectedness locus of these curves. First, we demonstrate that the curves have Tan-Lei similarity at Misiurewicz parameters, in a direct parallel with the Mandelbrot set. From there, we demonstrate that the boundary of the connectedness locus of sections of the curve can be understood as the quotient space of the period one case with respect to an equivalence relation induced by a certain critically periodic quadratic polynomial. We make progress towards the general conjecture, which would give a combinatorial description of the boundary of these periodic curves analogous to the quadratic minor lamination for the Mandelbrot set.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Recommended Citation
Young, Brady, "TOPOLOGICAL BOUNDARY PROPERTIES OF THE PERIODIC CUBIC PARAMETER CURVES" (2026). Open Access Dissertations. Paper 4555.
https://digitalcommons.uri.edu/oa_diss/4555