Lyapunov exponents and Hodge theory

Abstract

We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit integrals over moduli spaces of algebraic curves with additional structures. Moreover, these integrals can be interpreted as correlators in a topological string theory. Also a new analogy arose between ergodic theory and complex algebraic geometry.

16 pages, Latex, extended version of the talk by one of us (M.K.) at the conference "Mathematical Beauty of Physics", Saclay, June 1996, dedicated to the memory of C.Itzykson

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• Teichmueller curves generated by Weierstrass Prym eigenforms in genus three and genus four
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• Asymptotic behaviour of ergodic integrals of `renormalizable' parabolic flows
• A non-varying phenomenon with an application to the wind-tree model
• Self-inverses in Rauzy Classes
• Geodesics on Flat Surfaces
• Abelian Varieties, RCFTs, Attractors, and Hitchin Functional in Two Dimensions
• Ruelle resonances from cohomological equations
• Decay of Correlations for the Rauzy-Veech-Zorich Induction Map on the Space of Interval Exchange Transformations and the Central Limit Theorem for the Teichmueller Flow on the Moduli Space of Abelian Differentials
• Logarithmic asymptotics for the number of periodic orbits of the Teichmueller flow on Veech's space of zippered rectangles
• Instantons, Fluxons and Open Gauge String Theory
• Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula
• Effective Unique Ergodicity and Weak Mixing of Translation Flows
• Limit theorems for translation flows (published version)
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• Diagonal flow detects topology of strata
• A Central Limit Theorem for the Kontsevich-Zorich Cocycle
• The fine structure of Kontsevich-Zorich strata for genus 3