Last Thursday (September 23th), my friend Giovanni Forni uploaded to Arxiv his paper “A geometric criterion for the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle” with a short Appendix by me (after his kind invitation to contribute with some examples related to his results).

This paper of Giovanni’s grew out of some questions of the participants (in special A. Avila, S. Gouezel and R. Krikorian) of the “*Groupe de travail en Geometrie et Dynamique*” of Paris VI University (during the academic year 2003/2004) about G. Forni’s 2001 seminal paper. In his 2001 article (sometimes quoted as Giovanni Forni “*opus magnum*” by his friends due to the fact that it is a 103 pages long seminal paper), Giovanni studied *geometrical conditions* verified by the canonical (Masur-Veech) probability measure ensuring the *non-uniform hyperbolicity* of the *Kontsevich-Zorich (KZ) cocycle* over the *Teichmuller flow* (for a nice account on Forni’s 2001 paper, see this survey of W. Veech). On the other hand, since the KZ cocycle admit several others invariant probability measures, one could hope to follow Giovanni’s treatment of the Masur-Veech probability and try to study non-uniform hyperbolicity of KZ cocycle with respect to *other* invariant probability measures. However, the huge size of the paper makes it difficult to *isolate* the specific geometrical conditions (satisfied by the Masur-Veech measure) ensuring non-uniform hyperbolicity *in general*.

In fact, the main goal of Giovanni’s recent preprint is to provide sufficient conditions guaranteeing non-uniform hyperbolicity of the KZ cocycle with respect to “decent” invariant probabilities. The following result gives a prototype of Giovanni’s criterion:

Theorem(Theorem 4 of Giovanni’s paper).Consider the Kontsevich-Zorich cocycle over the support of a “decent” invariant probability measure (i.e., is assumed to be ergodic w.r.t. Teichmuller flow and -invariant, besides a technical condition calledlocal product structure).

Suppose that verifies the following geometrical condition: there is atranslation surface in the support of such that thehorizontal foliation (determined by the orbits of the horizontal direction) iscompletely periodic(i.e., the leaves of the foliations are simple closed curves or a segment joining two singularities of the foliation) and itshomological dimensionis (i.e., the subspace of the absolute homology generated by the homology classes of simple closed curves of the foliation has dimension equal to , where is the genus of the surface).

Then, the Kontsevich-Zorich cocyclewith respect tohas non-zero Lyapunov exponents (at least).

The standard reference for an excellent overview of the dynamics on the moduli space of translation surfaces (in particular, Teichmuller and dynamics and Kontsevich-Zorich cocycle) is Anton Zorich’s survey.

Notice that an immediate consequence of this result is the fact that if is *Lagrangian*, i.e., some translation surface in its support has *maximal* homological dimension , then the KZ cocycle over the support of is *non-uniformly hyperbolic* (i.e., all Lyapunov exponents are non-zero). For more explanations of these geometrical criteria, I strongly recommend reading directly Giovanni’s paper (since it is well-written and its applications to concrete examples are neat).

In any event, during the preparation of his manuscript, Giovanni and I talked about the issue of a geometrical criterion for the *simplicity* of the KZ cocycle, i.e., a sufficient condition to ensure that the Lyapunov exponents are all non-zero *and* they have multiplicity 1. In fact, this is a natural question because, despite the fact that simplicity is a stronger property than non-uniform hyperbolicity, A. Avila and M. Viana showed KZ cocycle *with respect to Masur-Veech measure* is *simple* (this was previously known as *Kontsevich-Zorich conjecture*). However, the method used by Avila and Viana relies on *combinatorial properties* of the so-called *Rauzy-Veech algorithm* (which is well-adapted to Masur-Veech measure, but unsuited for other invariant measures).

In particular, Giovanni and I suspected that his geometrical criteria wasn’t sufficient to obtain simplicity. Also, we were suspicious that the lower bound on the number of non-zero Lyapunov exponents provided by the homological dimension wasn’t sharp, and, in particular, we believed that the KZ cocycle could be non-uniformly hyperbolic *even *on the support of some *non-Lagrangian* measure! After some time, it turned out that certain special *square-tiled surfaces* are the desired “counter-examples” for the first *two* questions: firstly, by a double-cover construction, it is possible to construct a square-tiled surface with maximal homological dimension (i.e., Lagrangian) with non-simple spectrum (i.e., there is some multiple Lyapunov exponent), and, secondly, a certain family of square-tiled surfaces (parametrized by odd integers ) introduced by J. C. Yoccoz and I (in this paper here) gives examples of closed -invariant probabilities with *homological dimension 1* such that the number of non-zero Lyapunov exponents becomes *arbitrarily large* when . However, up to now, neither me nor Giovanni are aware of a non-uniformly hyperbolic *but* also non-Lagrangian measure (for instance, while the second family of examples have homological dimension and an arbitrarily large number of non-zero exponents, it is possible to check that there are also some zero exponents). In other words, the third question is still open. In any case, the details of the construction of these two examples is the content of the Appendix to Giovanni’s paper.

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