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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Functions nice in each variable separately

Consider the following question. If {f:{\mathbb R}^2\rightarrow {\mathbb R}} is continuous in each variable separately, need it be continuous in both? Although Cauchy (incorrectly) said it should, note it is not even obvious that it should be measurable.

A classical example is

\displaystyle  f(x,y):= \frac {xy}{x^2+y^2}

Defining {f(0,0)=0}, this function is real-analytic in each variable separately, but it is not even continuous at the origin.

By contrast, in the complex-analytic situation we have the following result.

Theorem 1 (Hartogs) Suppose {f:{\mathbb C}^{n_1}\times {\mathbb C}^{n_2}\rightarrow {\mathbb C}} is holomorphic in each variable separately. Then {f} is holomorphic on {{\mathbb C}^{n_1+n_2}}.

Below the fold, I will state some more results of this type and prove some of them. Continue reading