Carlos Matheus

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.

A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich–Zorich cocycle over the Teichmüller flow on the

Equilibrium states for random non-uniformly expanding maps

{\mathrm {SL}}_2(\mathbb {R})

Decidability of Chaos for Some Families of Dynamical Systems

SL2(R)-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M. Viana with respect to the so-called Masur–Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus

Rates of mixing for the Weil–Petersson geodesic flow I: No rapid mixing in non-exceptional moduli spaces

{\geqslant }3