Symplectic and isometric SL(2,#x211D;)-invariant subbundles of the Hodge bundle

Suppose N is an affine {\mathrm{SL}(2,{\mathbb{R}})} -invariant submanifold of the moduli space of pairs (M,\omega) , where M is a curve, and ω is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal {\mathrm{SL}(2,{\mathbb{R}})} -invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.

Lyapunov spectrum of invariant subbundles of the Hodge bundle

We study the Lyapunov spectrum of the Kontsevich–Zorich cocycle on SL(2,ℝ)-invariant subbundles of the Hodge bundle over the support of SL(2,ℝ)-invariant probability measures on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (the Kodaira–Spencer map) of the Hodge bundle with respect to the Gauss–Manin connection and investigate the relations between the central Oseledets subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.

Adapted metrics for dominated splittings

A Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A dominated splitting is a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of an adapted metric for a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub. Invariant Manifolds(Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.

Prevalence of non-Lipschitz Anosov foliations

We give sharp regularity results for the invariant subbundles of hyperbolic dynamical systems in terms of contraction and expansion rates and prove optimality in a strong sense: we construct open dense sets of codimension one systems where this regularity is not exceeded. Furthermore, we exhibit open dense sets of symplectic, geodesic, and codimension one systems where the analogous regularity results of [PSW] are optimal. As our main result we produce open sets of symplectic Anosov diffeomorphisms and flows with low transverse Hölder regularity of the invariant foliations almost everywhere. Prevalence of low regularity of conjugacies is a corollary. We also establish a new connection between the transverse regularity of foliations and their tangent subbundles.