The book intends to cover the most recent developments in hyperbolic dynamics and geometry making a systematic use of microlocal techniques, especially, of anisotropic spaces of distributions. It is still in construction and comments are welcome!
Latest version of the book (07/03/2023).
In a first part, we review the standard microlocal theory on manifolds. Based on that, we then develop the theory of Pollicott-Ruelle resonances for uniformly hyperbolic (Anosov) flows and derive the main consequences regarding its statistical properties. In a third part, we introduce the framework of Anosov Riemannian manifolds, namely, Riemannian spaces whose geodesic flow is Anosov, and show that all closed negatively-curved manifolds belong to this class. Eventually, we apply all these results to some classical problems in dynamics/geometry such as: the marked length spectrum conjecture, the ergodicity conjecture for the frame flow, etc.
– Lecture 1: Introduction, geometry/analysis on the unit tangent bundle.
– Lecture 2: Hyperbolic dynamics, linear rigidity of the marked length spectrum.
– Lecture 3: Microlocal analysis, spectral theory of Anosov flows.
– Lecture 4: Nonlinear rigidity of the marked length spectrum.