The book intends to cover recent developments in hyperbolic dynamics and geometry, making a systematic use of microlocal techniques. In a first part, we review the standard theory of microlocal analysis on closed manifolds. Based on that, we then develop the theory of Pollicott-Ruelle resonances for uniformly hyperbolic (Anosov) flows and derive important consequences regarding their 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. Finally, we apply all these results to some classical problems in dynamics/geometry such as: the marked length spectrum conjecture, ergodicity conjecture for frame flows, tensor tomography, etc.
Can be purchased on the SMF online bookshop here.
Recorded lectures: I gave four lectures at the Collège de France (Peccot lectures) on the content of the book.
– 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.
I gave three lectures in Northwestern on the content of Chapter 9 of the book (spectral theory of Anosov flows).
– Lecture 1: L^2 spectral theory of Anosov flows
– Lecture 2: Faure-Sjöstrand’s theorem. Pollicott-Ruelle resonances.
– Lecture 3: Application to ergodicity and mixing of volume-preserving Anosov flows.
Errata list:
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THIBAULT LEFEUVRE 