TCC Course - Fall 2010
Introduction to Teichmueller Dynamics
Dr. Corinna Ulcigrai (University of Bristol)
The study of dynamical properties of billiards in polygons, translation surfaces and piecewise isometries of the interval is a rich area of research which has developed and bloomed in the last decades. The main tool which allow to investigate these systems is given by the geodesic flow and the SL(2,R) action on the moduli spaces of translation surfaces.
During the lectures, we will introduce basic definitions and examples of:
- Riemann surfaces, moduli space of Riemann surfaces and Teichmueller space;
- Translations surfaces, Abelian and quadratic differentials and their moduli spaces, SL(2,R) action
- quasi-conformal maps, Teichmuller distance and Teichmuller geodesic flow.
We plan on proving during the course the following theorems, which show the beutiful connection between Teichmueller dynamics and dynamical properties of translation surfaces. Specifically, we will cover:
Additional topics will be covered time permitting.
- Unique ergodicity in almost every direction (Kenyon-Masur-Smillie);
- Quadratic asymptotics of saddle connections (Eskin-Masur);
- Veech dynamical dichotomy for lattice surfaces. (following Vorobetz);
Time: Wednesdays, 11.15am - 13.15am; See TCC Timetable
- Lecture 1 (Introduction/Outline of the course, Riemann surfaces and moduli space definitions)
- Lecture 2 (Moduli space, Teichmueller space, Mapping class groups: definitions and torus example, QC maps)
- Lecture 3 (Moduli space, Teichmueller distance, Teichmueller geodesic flow, translation and half-translation surfaces, Weyl theorem)
- Lecture 4 (Unique Ergodicity Part I: recurrent geodesics and Masur's Criterium)
- Lecture 5 (Unique Ergodicity Part II: non divergence)
- Lecture 6 (Further results on NUE(S) vs D(S) - Introduction to Counting Problems (holonomies, period coordinates, statemens of quadratic asymptotics)
- Lecture 7 (Quadratic asymptotics of closed curves and saddle connections (Siegel-Veech formula, proof of Eskin-Masur theorem)
- Lecture 8 (Veech Group, lattice surfaces properties, proof of Veech Dichotomy)
Course Assessment: The course will be assessed by solutions to the problems in the problem sheet posted here, that will be regularly updated as the course goes on (last updated: January, 2010).
- Genetators of SL(2,Z)
- A brief introduction to hyperbolic geometry can be found in:
- Chapter II of Bekka, Mayer "Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces", Cambdridge University Pres, 2000 (LMS Lecture Note Series 269).
- Chapter I, by Alan Beardon in "Ergodic theory, symbolic dynamics and hyperbolic spaces", edited by Bedford, Keane, Series, Oxford University Press 1991.
- A longer one, if the link is available by courtesy of Charles Walkden, is given in
"Lecture Notes on Hyperbolic Geometry" by Charles Walkden, University of Manchester
- Some basic ergodic theory notions can be found in:
- Chapter 4.1 of Katok, Hasselblatt "Introduction to the Modern Theory of Dynamical systems"
- Chapter I of Cornfeld, Fomin, Sinai "Ergodic Theory", Springer-Verlag, 1980.
- For the first part (basic objects, definitions and examples):
- Chapter 1 of "An introduction to Teichmueller spaces" by Y. Imayoshi and M. Taniguchi (Moduli space, Teichmueller space, quasi-conformal maps, Teichmueller distance)
- Appendix of the lecture notes "Dynamics of Interval exchange maps and Teichmueller geodesic flows" by Marcelo Viana (for a short summary containing all basic definitions)
- Howard Masur,"Ergodic theory of translation surfaces. In “Handbook of dynamical systems", Elsevier B. V., Amsterdam, 2006 (Sections 1.8, 2.1-4 for the definitions of translation surfaces, Abelian differentials, SL(2,R) action)
- For the second part (some theorems in Teichmueller dynamics):
- For Masur's Criterium and Kerckhoff, Masur, Smillie Theorem on Unique Ergodicity:
- Lecture Notes by Sebastien Gouezel, Erwan Lanneau on "Un theoreme de Kerkhoff, Masur et Smillie: Unique Ergodicite sur les surfaces plates", on Arxiv, in French
- Survey paper by Masur and Tabachnikov, Rational billiards and flat structures. Handbook of dynamical systems, Vol. 1A, 1015--1089, North-Holland, Amsterdam, 2002 (dvi without Figures avaiable to download on Masur's webpage ).
- S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Mathematics 124
- A. Eskin and H. Masur, Asymptotic formulas on flat surfaces., Ergodic Theory Dynam. Systems 21 (2001), no. 2, 443--478
- Yaroslav Vorobetz, Planar Structures and Billiards in Rational Polygons: the Veech alternative