3-dimensional Anosov flows: Exponential Mixing and Transfer Operators

This lecture explores the concept of 3-dimensional Anosov flows, focusing on exponential mixing and transfer operators. The instructor discusses the reduction to suspension of piecewise expanding maps, exponential mixing properties, and the conjecture of exponential mixing for equilibrium measures. Various examples and counterexamples are presented, including Ruelle's example and the Butterley-War theorem. The lecture also covers the doubling property, Gibbs property, and the construction of transfer operators. Quasi-compactness is introduced as a key concept to prove exponential mixing. The session concludes with the discussion of equilibrium measures and positive eigenfunctions. Theoretical results are illustrated with practical applications and mathematical proofs.