Connections Between Complex Dynamics, Statistical Physics, and Limiting
Spectra of Self-similar Group Actions
Indiana University Purdue University Indianapolis from August 15 to 19, 2016.
Funded by the National Science Foundation, the IUPUI School of Science, and the IUPUI Office for Academic Affairs.
Material for the learning talks.
The objective of these talks is for everyone to learn how certain problems from the Ising Model and self-similar group theories are related to iteration of rational maps (often in two variables).
For the most part, the learning talks will present sections of the following three papers (and of earlier works leading up to them):
[GZ] The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum (by Grigorchuk and Zuk)
[FJ] Brolin's Theorem for Curves in Two Complex Dimensions (by Favre and Jonsson)
[BLR] Lee-Yang Zeros for the DHL and 2D Rational Dynamics, I. Foliation of the Physical Cylinder. (by Bleher, Lyubich, and Roeder).
A good general reference is Baxter's Book (free online).
Good general references: Nekrashevich survey, Zuk survey.
Several Complex Variables Dynamics (and connections):
- Hongming Nei (Indiana University): Self-similar groups and Iterated Monodromy Groups.
Material through Section 3.1, skipping Example 2.5.4, of Nekrashevich survey.
- Yuki Takahashi (UC Irvine): Calculation of the spectrum for the action of the Lamplighter group.
Sections 6 and maybe 7 of [GZ].
- Scott Kaschner (Butler U): Rational map in two variables arising when trying to compute the spectrum of the action of the IMG for z^2-1.
This talk will be based on some calculations Scott, Rodrigo, and Roland did a few years ago. (No notes are available to post.)
Good general references are Paper by Hubbard and Papadopol, Sibony's survey and Dinh and Sibony's later survey.
Further Instructions For the Speakers:
We would like very much for these initial learning talks to do the following:
Preparing a good talk on your assigned topic may require some discussions with us by email or skype--we're all very friendly and open to discussion. Please contact us about whatever you need.
- provide a background for each of the topics generally
- provide background for specific connections between the topics
- have a colloquium-style atmosphere
- welcome questions and comments from the audience
- be accessible to a first-year graduate student
- be approximately forty minutes in length (and no more than fifty).