## Dissertations, Theses, and Capstone Projects

### Symmetric Rigidity for Circle Endomorphisms with Bounded Geometry and Their Dual Maps

6-2020

Dissertation

Ph.D.

Mathematics

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 $f=g$.

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