Constructing subdivision rules from rational maps

Subdivision rule constructions on critically preperiodic quadratic matings

Mating is an operation that identifies the domains of a polynomial pair in order to obtain a new map on the resulting quotient space. The dynamics of the mating are then dependent on the two polynomials and the manner in which the quotient space was defined, which can be difficult to visualize. This research addresses using Hubbard trees and finite subdivision rules as tools to examine quadratic matings with preperiodic critical points. In many cases, discrete parameter information on such quadratic pairs can be translated into topological information on the dynamics of their mating. The centr…

Constructing rational maps with cluster points using the mating operation

In this article, we show that all admissible rational maps with fixed or period two cluster cycles can be constructed by the mating of polynomials. We also investigate the polynomials which make up the matings that construct these rational maps. In the one cluster case, one of the polynomials must be an $n$-rabbit and in the two cluster case, one of the maps must be either $f$, a "double rabbit", or $g$, a secondary map which lies in the wake of the double rabbit $f$. There is also a very simple combinatorial way of classifiying the maps which must partner the aforementioned polynomials to cre…

Thurston's algorithm and rational maps from quadratic polynomial matings

Topological mating is an combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map $F$ on the Riemann sphere. Given a pair of polynomials of the form $z^2+c$ that are postcritically finite, there is a fast test on the constant parameters to determine whether this map $F$ exists---but this test is not constructive. We present an iterative method that utilizes finite subdivision rules and Thurston's algorithm to approximate this rational map, $F$. Thi…

具有Ahlfors-正则共形维数一的Julia集

Julia sets with Ahlfors-regular conformal dimension one

本文研究了后临界有限的超限有理映射的Julia集的Ahlfors-正则共形维数一的条件，证明其等价于该映射为crochet映射，即存在一个不变的连通图包含后临界集且其在图上的拓扑熵为零。

Surgery on postcritically finite rational maps by blowing up an arc

Using Thurston's characterization of postcritically finite rational functions as branched coverings of the sphere to itself, we give a new method of constructing new conformal dynamical systems out of old ones. Let $f(z)$ be a rational map and suppose that the postcritical set $P(f)$ is finite. Let $α$ be an embedded closed arc in the sphere and suppose that $f|α$ is a homeomorphism. Define a branched covering $g$ as follows. Cut the sphere open along $α$. Glue in a closed disc $D$. Map $S^{2} - \Int (D)$ via $f$ and $\Int (D)$ by a homeomorphism to the complement of $f(α)$. We prove theorems …