Dissertations, Theses, and Capstone Projects
Date of Degree
6-2020
Document Type
Dissertation
Degree Name
Ph.D.
Program
Mathematics
Advisor
Yunping Jiang
Committee Members
Frederick Gerdiner
Linda Keen
Sudeb Mitra
Zhe Wang
Yunchun Hu
Subject Categories
Analysis | Dynamical Systems
Abstract
Let $f$ be a circle endomorphism of degree $d\geq2$ that generates a sequence of Markov partitions that either has bounded nearby geometry and bounded geometry, or else just has bounded geometry, with respect to normalized Lebesgue measure. We define the dual symbolic space $\S^*$ and the dual circle endomorphism $f^*=\tilde{h}\circ f\circ{h}^{-1}$, which is topologically conjugate to $f$. We describe some properties of the topological conjugacy $\tilde{h}$. We also describe an algorithm for generating arbitrary circle endomorphisms $f$ with bounded geometry that preserve Lebesgue measure and their corresponding dual circle endomorphisms $f^*$ as well as the conjugacy $\tilde{h}$, and implement it using MATLAB.
We use the property of bounded geometry to define a convergent Martingale on $\S^*$, and apply the study of such Martingales to obtain a rigidity theorem. Suppose $f$ and $g$ are two circle endomorphisms of the same degree $d\geq 2$ such that each has bounded geometry and each preserves the normalized Lebesgue probability measure. Suppose that $f$ and $g$ are symmetrically conjugate. That is, $g=h\circ f\circ h^{-1}$ and $h$ is a symmetric circle homeomorphism. We define a property called locally constant limit of Martingale, and show that if $f$ has this property then $f=g$.
Recommended Citation
Adamski, John, "Symmetric Rigidity for Circle Endomorphisms with Bounded Geometry and Their Dual Maps" (2020). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/3790