Giovanni Forni

Invariant distributions and time averages for horocycle flows

There are infinitely many obstructions to existence of smooth solutions of the cohomological equation Uu = f , where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann M surface of finite area and f is a given function on SM . We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows.

SQUARE-TILED CYCLIC COVERS

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmuller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmuller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmuller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmuller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmuller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.

ZERO LYAPUNOV EXPONENTS OF THE HODGE BUNDLE

By the results of G. Forni and of R. Trevino, the Lyapunov spectrum of the Hodge bundle over the Teichmuller geodesic flow on the strata of Abelian and of qua- dratic differentials does not contain zeroes even though fo r certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss—Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichmuller curve, for which the real Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mech- anism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2,R)-invariant probability measure.

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.

DEVIATION OF ERGODIC AVERAGES FOR RATIONAL POLYGONAL BILLIARDS

We prove a polynomial upper bound on the deviation of er- godic averages for almost all directional flows on every tran slation sur- face, in particular, for the generic directional flow of bill iards in any Euclidean polygon with rational angles.

On the cohomological equation for nilflows

Let X be a vector field on a compact connected manifold M. An important question in dynamical systems is to know when a function g:M -> R is a coboundary for the flow generated by X, i.e. when there exists a function f: M->R such that Xf=g. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions D_n such that any sufficiently smooth function g is a coboundary iff it belongs to the kernel of all the distributions D_n.