5 research outputs found
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
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
-rabbit and in the two cluster case, one of the maps must be either , a
"double rabbit", or , a secondary map which lies in the wake of the double
rabbit . 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