arXiv subjects

Thomas Sharland

Publications and source records attributed to Thomas Sharland.

7 recordsLinked to original sources

arXiv2023

On the deck groups of iterates of bicritical rational maps

Given a rational map $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ on the Riemann sphere, we define $\mathrm{Deck}(f)$ to be the group of Möbius transformations $μ$ satisfying $f \circ μ= f$. In this note, we consider the groups $\mathrm{Deck}(f^k)$, where $f$ is a \emph{bicritical} rational map (that is, a rational map with exactly two critical points) and $f^k$ denotes the $k$th iterate of $f$. In particular, we give a complete description of which groups (up to isomorphism) arise as the groups $\mathrm{Deck}(f^k)$ for bicritical rational maps $f$.

math.DS↗

arXiv2023

Matings of cubic polynomials with a fixed critical point. Part II: $α$-symmetry of limbs

In this article we provide a combinatorial sufficient (and conjecturally, necessary) condition (called $α$-symmetry) for the mating of two postcritically finite polynomials in $\mathcal{S}_1$ to be obstructed. To do this, we study the rotation sets associated to the parameter limbs in the connectedness locus of $\mathcal{S}_1$, which allows us to determine when there exist ray classes in the formal mating which contain a closed loop. We give a proof of the necessity of $α$-symmetry for a particular subset of postcritically finite maps in $\mathcal{S}_1$. Many examples are given to illustrate the results of the paper.

math.DS↗

arXiv2022

Bicritical rational maps with a common iterate

Let $f$ be a degree $d$ bicritical rational map with critical point set $\mathcal{C}_f$ and critical value set $\mathcal{V}_f$. Using the group $\textrm{Deck}(f^k)$ of deck transformations of $f^k$, we show that if $g$ is a bicritical rational map which shares an iterate with $f$ then $\mathcal{C}_f = \mathcal{C}_g$ and $\mathcal{V}_f = \mathcal{V}_g$. Using this, we show that if two bicritical rational maps of even degree $d$ share an iterate then they share a second iterate, and both maps belong to the symmetry locus of degree $d$ bicritical rational maps.

math.DS↗

arXiv2022

Matings of Cubic polynomials with a fixed critical point, Part I: Thurston Obstructions

We prove that if $F$ is a degree $3$ Thurston map with two fixed critical points, then any irreducible obstruction for $F$ contains a Levy cycle. As a corollary, it will be shown that if $f$ and $g$ are two postcritically finite cubic polynomials each having a fixed critical point, then any obstruction to the mating $f \mate g$ contains a Levy cycle. We end with an appendix to show examples of the obstructions described in the paper.

math.DS↗

arXiv2022

Relations between Escape Regions in the Parameter Space of Cubic Polynomials

We describe a topological relationship between slices of the parameter space of cubic maps. In the paper \cite{CP1}, Milnor defined the curves $\mathcal{S}_n$ as the set of all cubic polynomials with a marked critical point of period $p$. In this paper, we will describe a relationship between the boundaries of the connectedness loci in the curves $\mathcal{S}_1$ and $\mathcal{S}_2$.

math.DS↗

arXiv2011

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 create rational maps with cluster cycles. Finally, we also investigate the multiplicities of the shared matings arising from the matings in the paper.

math.DS↗

arXiv2011

Thurston equivalence for rational maps with clusters

We investigate rational maps with period one and two cluster cycles. Given the definition of a cluster, we show that, in the case where the degree is $d$ and the cluster is fixed, the Thurston class of a rational map is fixed by the combinatorial rotation number $ρ$ and the critical displacement $δ$ of the cluster cycle. The same result will also be proved in the case that the rational map is quadratic and has a period two cluster cycle, but that the statement is no longer true in the higher degree case.

math.DS↗