A Fundamental Domain for V_3

TL;DR

This paper constructs a fundamental domain for a specific punctured Riemann surface related to quadratic rational maps with critical points, providing insights into the dynamics and hyperbolic components of the parameter space.

Contribution

It introduces a fundamental domain for the surface $V_{3,m}$, offering a topological description of dynamics in type III hyperbolic components of $V_3$.

Findings

Description of the fundamental domain for $V_{3,m}$

Topological conjugacy classification of dynamics in hyperbolic components

Insights into the structure of the parameter space $V_3$

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.