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.Comment: 23 page

Thurston equivalence for rational maps with clusters

We investigate rational maps with period-one and period-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 δof the cluster cycle. The same result will also be proved in the case where the rational map is quadratic and has a period-two cluster cycle, and we will also show that the statement is no longer true in the higher-degree case

Mating Siegel and parabolic quadratic polynomials

Let $f_\theta(z)=e^{2\pi i\theta}z+z^2$ be the quadratic polynomial having an indifferent fixed point at the origin. For any bounded type irrational number $\theta\in\mathbb{R}\setminus\mathbb{Q}$ and any rational number $\nu\in\mathbb{Q}$, we prove that $f_\theta$ and $f_\nu$ are conformally mateable, and that the mating is unique up to conjugacy by a M\"{o}bius map. This gives an affirmative (partial) answer to a question raised by Milnor in 2004. A crucial ingredient in the proof relies on an expansive property when iterating certain rational maps near Siegel disk boundaries. Combining this with the expanding property in repelling petals of parabolic points, we also prove that the Julia sets of a class of Siegel rational maps with parabolic points are locally connected.Comment: 40 pages, 9 figure

Rational maps with clustering and the mating of polynomials

The main focus of this thesis is the study of a special class of bicritical rational maps of the Riemann sphere. This special property will be called clustering; which informally is when a subcollection of the immediate basins of the two (super-)attracting periodic orbits meet at a periodic point p, and so the basins of the attracting periodic orbits are clustered around the points on the orbit of p. Restricting ourselves to the cases where p is fixed or of period 2, we investigate the structure of such maps combinatorially; in particular showing a very simple collection of combinatorial data is enough to define a rational map uniquely in the sense of Thurston. We also use the language of symbolic dynamics to investigate pairs (f, g) of polynomials such that f - g has a fixed or period two cluster point. We find that that the internal addresses of such maps follow very definite patterns which can be shown to hold in general.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (EPSRC)GBUnited Kingdo