It was proved by Dolgopyat and Pesin that any compact smooth Riemannian manifold admits ergodic volume-preserving smooth diffeomorphisms. I will discuss the following question (due to Katok): What are the possible Lyapunov spectra (with respect to the volume measure) of ergodic diffeomorphisms? The strategy to answer the question is to begin with a diffeomorphism whose exponents are far apart and them mix them carefully using a deformation of Baraviera-Bonatti type. I will explain how to implement this strategy in the case of Anosov diffeomorphisms. Another question is: Which are the possible Lyapunov spectra in a fixed homotopy class of volume-preserving Anosov diffeos? I will discuss the possibility of an "exotic" Anosov example. This talk is based on joint work with A. Katok and F. Rodriguez Hertz.
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