## 61 Citations

A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle

- Mathematics
- 2010

We prove a geometric criterion on a $\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the…

Zero Lyapunov exponents of the Hodge bundle

- Mathematics
- 2012

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…

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

- Mathematics
- 2015

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich–Zorich cocycle over the Teichmüller flow on the $${\mathrm {SL}}_2(\mathbb {R})$$SL2(R)-orbit…

Quantitative behavior of non-integrable systems. I

- Mathematics
- 2020

The theory of Uniform Distribution started with the equidistribution of the irrational rotation of the circle, proved around 1905 independently by Bohl, Sierpinski and Weyl. The quantitative…

A coding-free simplicity criterion for the Lyapunov exponents of Teichmüller curves

- Mathematics
- 2012

In this note we show that the results of Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the…

Introduction to Teichm\"uller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

- Mathematics
- 2013

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school "Modern Dynamics and its Interaction with…

Lyapunov spectrum of invariant subbundles of the Hodge bundle

- MathematicsErgodic Theory and Dynamical Systems
- 2012

Abstract 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…

Teichmuller curves, triangle groups, and Lyapunov exponents

- Mathematics
- 2005

We construct a Teichmuller curve uniformized by a Fuchsian triangle group commensurable to �(m, n, ∞) for every m, n ≤ ∞. In most cases, for example when m 6 n and m or n is odd, the uniformizing…

Affine invariant submanifolds with completely degenerate Kontsevich–Zorich spectrum

- MathematicsErgodic Theory and Dynamical Systems
- 2016

We prove that if the Lyapunov spectrum of the Kontsevich–Zorich cocycle over an affine $\text{SL}_{2}(\mathbb{R})$ -invariant submanifold is completely degenerate, i.e. if…

SQUARE-TILED CYCLIC COVERS

- Mathematics
- 2011

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,…

## References

SHOWING 1-10 OF 48 REFERENCES

Deviation of ergodic averages for area-preserving flows on surfaces of higher genus

- Mathematics
- 2002

We prove a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on…

Asymptotic Behaviour of Ergodic Integrals of 'Renormalizab le' Parabolic Flows

- Mathematics
- 2002

Ten years ago A. Zorich discovered, by computer experiments on interval exchange transformations, some striking new power laws for the ergodic in tegrals of generic non-exact Hamiltonian flows on…

Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture

- Mathematics
- 2005

We prove the Zorich–Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichmüller ow on (any connected component of a stratum of) the moduli space of Abelian differentials on…

Asymptotic flag of an orientable measured foliation

- Mathematics
- 1993

We state several conjectures on asymptotic "spectral properties" of transformation operators involved in Rauzy induction for a generic interval exchange transformation. Modulo these conjectures we…

SOLUTIONS OF THE COHOMOLOGICAL EQUATION FOR AREA-PRESERVING FLOWS ON COMPACT SURFACES OF HIGHER GENUS

- Mathematics
- 1997

Let f be a local homeomorphism of the plane with a fixed point z which is a locally maximal invariant set and which is neither a sink nor a source. We prove that there are two integers q > 1 and r >…

Siegel measures

- Mathematics
- 1998

The goals of this paper are first to describe and then to apply an ergodictheoretic generalization of the Siegel integral formula from the geometry of numbers. The general formula will be seen to…

Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants

- Mathematics
- 2002

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment…

Moduli spaces of quadratic differentials

- Mathematics
- 1990

The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural…

On the geometry and dynamics of diffeomorphisms of surfaces

- Mathematics
- 1988

This article was widely circulated as a preprint, about 12 years ago. At that time the Bulletin did not accept research announcements, and after a couple of attempts to publish it, I gave up, and the…