A Fundamental Domain for V_3

Abstract

We describe a fundamental domain for the punctured Riemann surface $V_{3,m}$ which parametrises (up to M\"obius conjugacy) the set of quadratic rational maps with numbered critical points, such that the first critical point has period three, and such that the second critical point is not mapped in $m$ iterates or less to the periodic orbit of the first. This gives, in turn, a description, up to topological conjugacy, of all dynamics in all type III hyperbolic components in $V_{3}$, and gives indications of a topological model for $V_{3}$, together with the hyperbolic components contained in it.