Chapter 8 - On the Lyapunov Exponents of the Kontsevich–Zorich Cocycle

  title={Chapter 8 - On the Lyapunov Exponents of the Kontsevich–Zorich Cocycle},
  author={Giovanni Forni},
A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle
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
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
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
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
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
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
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
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
  • D. Aulicino
  • Mathematics
    Ergodic 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
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,


Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
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
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
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
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
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
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
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
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
Chapter 6 - An Introduction to Veech Surfaces
On the geometry and dynamics of diffeomorphisms of surfaces
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