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

By Artur Avila, Marcelo Viana. 2005

189 citations recorded by the source corpus.

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 compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d09ee112b5c4577ea1f1ae6e2d0992e06522943a

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Citations in the corpus

A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces (2013, methodology): “This conjecture is nowadays known to be true after the celebrated works of Forni [20], who proved that λMV g > 0, and Avila and Viana [4], who established the full conjecture.”
Extremal Lyapunov exponents: an invariance principle and applications (2010, result): “Assuming the base dynamics (g, μ) is fairly “chaotic” (hyperbolic, possibly in a non-uniform fashion), there is now a good understanding of such issues as the existence of non-zero Lyapunov exponents (see Bonatti, Gomez-Mont, Viana [14, 43]) or the simplicity of the Lyapunov spectrum (see Avila, Bon”
Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials (2007, methodology): “Avila and Viana used combinatorics of Rauzy-Veech induction to prove the simplicity of the essential part of the Lyapunov spectrum of the Teichmuller geodesic flow on the strata of Abelian differentials (see [ AV07 ]).”
Ergodic Theory of Interval Exchange Maps (2006, methodology): “Then, Avila and the present author [4, 3] proved the Zorich-Kontsevich [14, 25, 26] conjecture on the Lyapunov spectrum of the Teichmüller, following important partial results by Forni [8].”
Exponential mixing for the Teichmüller flow (2005, background): “For instance, the proof of typical weak mixing in [ AF ] can be made more transparent using such estimates.”
Limit theorems for translation flows (2008, methodology): “While, by the Avila-Viana Theorem [4], the Lya- punov spectrum of the Masur-Veech measure is simple, there are also natural examples of invariant measures with nonsimple positive second Lyapunov ex- ponent due to Eskin-Kontsevich-Zorich [17], G. Forni [22], and C. Matheus (see [22, App.”