Measure theoretic properties of rational maps missing period two orbits
- When?
- Friday 1 June 2012, 16:00 to 17:00
- Where?
- 22AA04
- Open to:
- Staff, Students
- Speaker:
- Professor Jane Hawkins (University of North Carolina)
Abstract: Baker posed a question in the 1960's about when a rational map of degree d can be lacking periodic points of (minimum) period k. He gave the short list of pairs (d,k) that can occur. In this talk we discuss the complete solution to this problem proved by Hawkins' Ph.D. student Rika Hagihara.
This problem is mentioned in Beardon's book Iteration of Rational Functions, and other earlier results include a paper by Kisaka suggesting the solution without proof, and a discussion by Milnor of the quadratic case. Hagihara showed that several of the cases involve holomorphic families of maps. We focus on recent developments, in particular we connect some of these maps to infinite measure preserving ergodic tranformations and inner functions. The Krengel entropy for a real parameter family of examples has been computed by Hawkins' student Rachel Bayless; we mention extensions to conformal measures in the general case.