Two weeks ago I attended a superb conference at MFO (Oberwolfach, Germany) entitled Billiards, Flat Surfaces and Dynamics on Moduli Spaces (organized by Howard Masur, Martin Möller and Anton Zorich). In particular, I gave a short talk on a work in progress with Giovanni Forni and Anton Zorich about new examples of -invariant loci of the moduli space of Abelian differentials supporting -invariant probabilities where the vanishing Lyapunov exponents of the Kontsevich-Zorich cocycle are not explained by geometrical means, in contrast with previous examples (square-tiled cyclic covers) where we knew that the vanishing Lyapunov exponents were geometrically “justified” from the fact that they came from the annihilator of the second fundamental form (Kodaira-Spencer map) of the Gauss-Manin connection on the Hodge bundle over the corresponding loci. In particular, in previous examples, we deduced that the neutral Oseledets bundle (i.e., the Oseledets subbundle associated to vanishing exponents) depended in a real-analytic way on the base point, a fact that is far from being true for general cocycles (and general invariant probabilities).
Of course, I plan to discuss this work with G. Forni and A. Zorich in the future (when we finish a preprint version of it), but since I believe it will take a while until we complete all details, I decided to make publicly available (for the interested reader) this expanded abstract of my talk (intended for the proceedings of the conference). Here I make a quick review of the action on the moduli space of Abelian differentials (more or less along the lines of these posts here) and then I sketch the description of the example and the fact that the neutral Oseledets bundles can be distinguished from the annihilator of the Kodaira-Spencer map by computing the homological action of appropriate pseudo-Anosovs (periodic orbits of Teichmuller flow) inside our locus.
[Update May 28th 2011: In the extended abstract of my talk, I made a side comment on a possible “name” for the Teichmuller curve in genus 4 with totally degenerate spectrum I found with G. Forni some time ago (e.g., see this blog post). In fact, Matt Bainbridge proposed to some people at the conference to think about a name for this object to contrast with the Eierlegende Wollmilchsau (which is the “official name” of the genus 3 Teichmuller curve with totally degenerate spectrum discovered by G. Forni and M. Moller, F. Herrlich and G. Schmithusen independently), and during last week I got a proposal from Barak Weiss and Vincent Delecroix: it should be called “Ornithorynque” (i.e., Platypus in French) because this is a very strange animal like the “Eierlegende Wollmilchsau”. Of course, while this sounds a nice suggestion to me, we’ll have to wait to see if the Teichmuller curves community is willing to adopt this name or not!😀 ]