Ruelle resonances for geodesic flow

This lecture by the instructor focuses on explaining the concept of Ruelle resonances for geodesic flows on non-compact manifolds. The lecture starts by discussing resonances for matrices and then delves into the more complex systems. The instructor explains the notion of Ruelle resonances through the lens of transfer operators and generating series, emphasizing the importance of smoothness in functions. The lecture also touches upon the asymptotic behavior of geodesic flows and the role of eigenvalues in the dynamics. The concept of Ruelle resonances is explored in the context of expanding maps on the circle, highlighting the significance of smoothness and the spectrum of the transfer operator. The lecture concludes with a discussion on the spectrum of the transfer operator acting on CR functions and the implications of the absence of resonances in certain maps.