The parameter capturemap for 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.