Remarks on quadratic rational maps

TL;DR

This paper provides an expository overview of quadratic rational maps, focusing on their geometry, topology, and dynamics, including moduli space exploration and real map theory, with minimal proofs and extensive technical appendices.

Contribution

It offers a comprehensive survey of quadratic rational maps, including geometric, topological, and dynamical aspects, with new insights into moduli space slices and real quadratic maps.

Findings

Exploration of moduli space via complex slices

Description of real quadratic rational maps

Analysis of Julia sets and their topological properties

Abstract

This will is an expository description of quadratic rational maps. Sections 2 through 6 are concerned with the geometry and topology of such maps. Sections 7--10 survey of some topics from the dynamics of quadratic rational maps. There are few proofs. Section 9 attempts to explore and picture moduli space by means of complex one-dimensional slices. Section 10 describes the theory of real quadratic rational maps. For convenience in exposition, some technical details have been relegated to appendices: Appendix A outlines some classical algebra. Appendix B describes the topology of the space of rational maps of degree \[d\]. Appendix C outlines several convenient normal forms for quadratic rational maps, and computes relations between various invariants.\break Appendix D describes some geometry associated with the curves \[\Per_n(\mu)\subset\M\]. Appendix E describes totally disconnected Julia sets containing no critical points. Finally, Appendix F, written in collaboration with Tan Lei, describes an example of a connected quadratic Julia set for which no two components of the complement have a common boundary point.