Entropy and the localization of eigenfunctions on negatively curved manifolds - I

Entropy and the localization of eigenfunctions on negatively curved manifolds - I

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Nalini Anantharaman , IAS
Fine Hall 601

We are interested in the behaviour of laplacian eigenfunctions on negatively curved manifolds, in the high frequency limit. The Quantum Unique Ergodicity conjecture predicts that they should become uniformly distributed over phase space, and the Shnirelman theorem states that this is true if we allow ourselves to possibly drop a ``negligible'' family of eigenfunctions. Nonnenmacher and I proved that, in any case, the eigenfunctions must in the high frequency regime have a large Kolmogorov-Sinai entropy : this prevents them, for instance, from concentrating on periodic geodesics. The proof uses notions from ergodic theory (such as entropy) mixed with techniques from linear PDE. Talk I : I will state the result, introduce some notions of microlocal analysis, recall the definition of entropy and state a technical estimate which is the core of the proof.