The parameter capturemap for V_3

TL;DR

This paper investigates the Wittner capture construction for certain critically finite quadratic rational maps, analyzing the number of representations in a specific parameter space and establishing density bounds for these representations.

Contribution

It introduces the parameter space V_3 and proves that the set of maps with a fixed number of Wittner capture representations is dense and bounded away from zero.

Findings

The set of maps with exactly 2^r representations is dense in V_3.

The measure of maps with a fixed number of representations is bounded away from zero.

The study characterizes the structure of representations in the parameter space V_3.

Abstract

This is a study of the Wittner capture construction for critically finite quadratic rational maps for which one critical point is periodic, and the second critical point is in the backward orbit of the first. This construction gives a way of describing rational maps up to topological conjugacy. It is known that representations as Wittner captures are not unique. We show that, in a certain parameter space which we call $V_3$, the set of maps with exactly $2^r$ representations as a Wittner capture is of density bounded from 0 for each $r\geq 0$, and for each fixed preperiod of the second critical point.