*Problems in Holomorphic Dynamics*

This preprint will be published by Springer-Verlag as a chapter of *Linear and Complex Analysis Problem Book* (eds. V. P. Havin and N. K. Nikolskii).

1. Quasiconformal Surgery and Deformations

- Ben Bielefeld: Questions in Quasiconformal Surgery
- Curt McMullen: Rational maps and Teichmüller space
- John Milnor: Problem: Thurston's algorithm without critical finiteness
- Mary Rees: A Possible Approach to a Complex Renormalization Problem

2. Geometry of Julia Sets

- Lennart Carleson: Geometry of Julia sets.
- John Milnor: Problems on local connectivity

3. Measurable Dynamics

- Mikhail Lyubich: Measure and Dimension of Julia Sets.
- Feliks Przytycki: On Invariant Measures for Iterations of Holomorphic Maps

4. Iterates of Entire Functions

- Robert Devaney: Open Questions in Non-Rational Complex Dynamics
- A. Eremenko and M. Lyubich: Wandering Domains for Holomorphic Maps

5. Newton's Method

- Scott Sutherland: Bad Polynomials for Newton's Method

*Distortion Results and Invariant Cantor Sets of Unimodal Maps*

A distortion theory is developed for $S-$unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of $S-$unimodal maps is classified according to a distortion property, called the Markov-property.

*Remarks on Quadratic Rational Maps*

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.
- 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.
- 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.