Exponential Mixing and Transfer Operators

In November 2024, I gave a talk at the final presentation session for the course Topics in Topology (Ergodic Theory with a view towards Number Theory) by Prof. Seonhee Lim. I introduced the transfer operator framework for establishing the exponential mixing properties of dynamical systems. In particular, I covered proofs of exponential mixing of one-dimensional expanding maps, and geodesic flows on compact surfaces of variable negative curvature. The talk was based on Pollicott’s lecture notes [6].

Abstract

Transfer operators offer a powerful framework for establishing the exponential mixing properties of dynamical systems and, in certain cases, provide the only available approach. In this talk, we explore the methodology for proving exponential mixing through the spectral analysis of transfer operators, highlighting the key concepts and techniques involved.

Slides

These are the slides for my talk.

References

[1] Bowen, R. and C. Series, Markov maps associated with Fuchsian groups, Publications Mathématiques de l’IHÉS 50 (1979), 153–170.

[2] Boyarsky, A. and P. Gora, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Birkhäuser, 1997.

[3] Dolgopyat, D., On decay of correlations in Anosov flows, Annals of Mathematics 2 (1998), 357–390.

[4] Galatolo, S., Statistical properties of dynamics: Introduction to the functional analytic approach, arXiv preprint (2015), arXiv:1510.02615

[5] Katok, S., and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bulletin of the American Mathematical Society 44 (2007), 87–132.

[6] Pollicott, M., Exponential Mixing: Lectures from Mumbai, Geometric and Ergodic Aspects of Group Actions. Springer, 2020. 135–167.

[7] Walkden, C., MAGIC: Ergodic Theory Lecture 9 - Thermodynamic Formalism, Electronic Edition, 2013. (Online)