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

By G. Forni. 2004

294 citations recorded by the source corpus.

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 higher genus surfaces. The result on the Teichmuller flow is formulated in terms of a (symplectic) cocycle on the real cohomology bundle over the moduli space of holomorphic differentials introduced by Kontsevich and Zorich. We prove that such a cocycle is non-uniformly hyperbolic, that is, all of its Lyapunov exponents are different from zero. In particular, the number of strictly positive exponents is equal to the genus of the surface. From this theorem we derive that ergodic integrals of smooth functions for generic area-preserving flows on higher genus surfaces grow with time according to a power-law asymptotics with a number of terms equal to the genus of the surface and stricltly positive exponents equal to the non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle. In particular, for conservative flows on surfaces of higher genus, the deviation of ergodic averages for a generic smooth function obeys a power law with a strictly positive exponent and, consequently, the Denjoy-Koksma inequality does not hold. The derivation of the deviation theorem relies in a fundamental way on the notion of invariant distribution for flows on surfaces and the related notion of basic current for the orbit foliation.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1977f855fa3cd408d87e65c42c40e990ea3d486e

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Citations in the corpus

Limit theorems for translation flows (2008, methodology): “(Recall that, as Veech and Forni showed, the first one, θ1 = 1, is always simple [41], [21] and that, by the Avila-Viana theorem [4], the second one is simple for the Masur- Veech smooth measure.)”
Beyond substitutive dynamical systems: S-adic expansions (2013, background): “It was then further generalized in [71] and [50] (see also [62]).”
Splitting mixed Hodge structures over affine invariant manifolds (2013, background): “(For H 1 , this is due to Forni [For02]; for W 0 , we can assume the cocycle is trivial.)”
Hölder Regularity of the Solutions of the Cohomological Equation for Roth Type Interval Exchange Maps (2004, background): “The ergodic properties of the continued fraction map and of the related Teichm¨ uller flow (see Section 4.2 for its definition) have been studied in detail [V5, V6, V7, Z2, Z4, Fo2 ].”
Ergodic Theory of Interval Exchange Maps (2006, methodology): “Then, Avila and the present author [3, 4] proved the Zorich-Kontsevich [14, 25, 26] conjecture on the Lyapunov spectrum of the Teichmüller, following important partial results by Forni [8].”
Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow (2010, methodology): “To prove this statement, let λ 0 , . . . , λ d u be the Lyapunov exponents of the Kontsevich-Zorich cocycle (see [For02]) restricted to the bundle ˜ E u , for the measure µ .”
Quantitative recurrence and large deviations for Teichmüller geodesic flow (2005, background): “Veech [26] showed that it was ‘measurably Anosov’, and more recently Forni [11] has obtained explicit formulas regarding the hyperbolic behavior.”
Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differentials (2002, background): “Recently G. Forni has found a proof that all �i, 1 ≤ i ≤ g are strictly positive [ 3 ].”
Chapter 8 - On the Lyapunov Exponents of the Kontsevich–Zorich Cocycle (2006, methodology): “The full Zorich conjecture, which affirms that the Lyapunov spectrum is simple, that is, all Lyapunov exponents are distinct, was left open in [21] in genus higher than 3.”