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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Large Deviations – Cramer’s theorem

Last summer I attended a conference on “Dynamics and Numbers” at the Max Planck Institute in Bonn. Vincent Delecroix gave a nice talk on large deviations for the Teichmuller flow, and I decided to learn a bit about this concept. I wrote the notes below at that time.

In this post, I will prove the simplest example of large deviations – Cramer’s theorem. Here is a simple example. Consider a random coin toss, and at each step you either win or loose {1} dollar. After {N} steps, consider the probability of having more than {aN} dollars, where {a>0} is fixed. Of course, it goes to zero as {N{\rightarrow} \infty}, but how fast? Playing around with Stirling’s formula reveals that the probability decays exponentially fast.

The theory of large deviations is concerned with these kinds of exponentially unlikely events. We now move on to a more precise setup. Continue reading

Entropy and set projections

Let {A\subset {\mathbb Z}^3} be a finite set, and let {A_{xy}, A_{yz}, A_{xz} } be its projections to the corresponding two-planes. Denoting by {\# S} the cardinality of a set {S}, we have the following inequality:

\displaystyle  \left(\# A\right)^2 \leq \left(\# A_{xy}\right)\left(\# A_{yz}\right) \left(\# A_{xy}\right)

This generalizes to sets in higher dimensions and projections to subspaces of possibly different dimensions. As long as each coordinate appears at least {n} times on the right, the size of {\left(\# A\right)^n} is bounded by the product of the projections.

One proof of the above inequality is via a mixed version of Cauchy-Schwartz or Holder-type inequalities. For {f,g,h} functions of two variables, we have

\displaystyle  \begin{array}{rcl}  \left(\int \int \int f\left(x,y\right) g\left(y,z\right) h\left(x,z\right) dx\, dy\, dz\right)^2 \leq\\ \leq\left(\int \int f\left(x,y\right)dx\, dy\right) \cdot \left(\int \int g\left(y,z\right)dy\, dz\right) \left(\int \int h\left(x,z\right)dx \, dz\right) \end{array}

Taking {f,g,h} to be indicators of projections of {A} to the {2}-planes, the bound follows. However, proving the inequality for {f,g,h} requires some work.

There is a different inequality, involving entropy of random variables, whose proof (and generalizations) are much more conceptual. Namely, if {X,Y,Z} are random variables and {H\left(-\right)} denotes entropy, we have

\displaystyle  2H\left(X,Y,Z\right) \leq H\left(X,Y\right) + H\left(Y,Z\right) + H\left(X,Z\right)

Below the fold are the definitions, a proof, and an application. Continue reading

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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