Quantum Ergodicity

The Quantum Ergodicity Theorem of Shnirelman, Colin de Verdière and Zelditch is a result that combines in a nice way several different concepts. Namely, suppose {(M,g)} is a compact Riemannian manifold with ergodic geodesic flow. Let {\{\phi_k\}} be the appropriately normalized eigenfunctions of the Laplacian {\Delta}. Then the measures with density {\frac 1 N\sum_{k=1}^N |\phi_k|^2 \cdot d{\mathrm{Vol}}} tend weakly to the usual volume {d{\mathrm{Vol}}}.

I find the result interesting because it makes a dynamical assumption on the metric and says something about the eigenfunctions of the Laplacian. The tool connecting these two aspects is microlocal analysis. It appears in the proof, but not in the statement of the result.

Below the fold is a brief discussion of the main ideas that go into the proof. Weyl’s law for the density of eigenvalues appears as a rather simple result of the general formalism of microlocal analysis.

I am following a series of lectures notes due to Faure and Anantharaman available here.

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Torsion on Jacobians and Teichmuller dynamics

In this post, I will describe a special case of a theorem about torsion points on Jacobians and Teichmuller dynamics which is proved in this paper. Namely, I will describe a different proof of a theorem of Moeller that for Teichmuller curves, zeroes of the holomorphic {1}-form must land in the torsion of (a factor of) the Jacobian. Some familiarity with Teichmuller dynamics will be assumed, as in the survey of Zorich.

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Semisimplicity of the Kontsevich-Zorich cocycle

In this post, I would like to discuss some of the ideas from my paper Semisimplicity and Rigidity of the Kontsevich-Zorich cocycle. It contains a number of somewhat disparate methods – Hodge theory, random walks and a bit of homogeneous and smooth dynamics. I will try to explain how these concepts interact and what the essential aspects are. Some familiarity with flat surfaces, as explained for example in the survey by Anton Zorich, will be assumed.

Below the fold I will discuss some of the analytic ideas from variations of Hodge structures. Then I’ll try to connect these to Teichmuller dynamics and explain how this leads to various semisimplicity results. For applications, I’ll discuss how this implies that measurable {{\mathrm{SL}}_2{\mathbb R}}-invariant bundles have to be real-analytic. I’ll also mention how real multiplication on factors of the Jacobians arises.

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Rigidity for Isometric Actions

In this post, I will discuss some elementary results about the action of an isometry on a compact metric space. I found them as homework problems assigned by Prof. Amie Wilkinson in a Smooth Dynamics course, and although quite elementary, I found the results rather striking.

In summary, (ergodic) actions by isometries look a lot like translations on compact abelian groups. For example, orbit closures in a Riemannian manifold are always tori. Precise statements are below the fold. Continue reading