Mihajlo Cekić : Speed of mixing for extensions of chaotic (Anosov) flows

Abstract: We will discuss the speed of mixing of extensions of chaotic (Anosov) flows to possibly non-compact principal bundles. For example, this includes the frame flow on the bundle of orthonormal frames over a negatively curved Riemannian manifold, and our result in this setting guarantees that ergodicity of the frame flow implies its rapid mixing, that is, mixing faster than C_N t^{-N} for any N > 0 (here t > 0 denotes time and C_N > 0). In the second situation of interest, we will consider extensions to Z^d covers; in this case, we will prove the optimal speed of mixing Ct^{-d/2}, and moreover we will provide the full asymptotic expansion in decaying integer powers of the time t. To prove these results we use the Fourier expansion in vertical fibers, suitably constructed anisotropic Sobolev spaces tailored to the dynamics, and a novel semiclassical calculus on principal G-bundles that we call the Borel-Weil calculus, where the semiclassical parameters correspond to the highest roots parametrizing irreducible representations of G. The calculus can be shown to have further applications to hypoellipticity and quantum ergodicity of horizontal (sub-)Laplacians. Joint works with Thibault Lefeuvre and Sebastián Muñoz Thon.