Lyapunov exponents and Hodge theory

TL;DR

This paper explores the connection between Lyapunov exponents in ergodic dynamical systems and Hodge theory, revealing new formulas linking dynamical invariants with algebraic geometry and topological string theory.

Contribution

It introduces a novel formula relating characteristic exponents to integrals over moduli spaces, bridging ergodic theory and algebraic geometry.

Findings

Correlators decay as a power law with exponent 1/3 in simple systems

Derived explicit integral formulas connecting exponents with moduli space geometry

Established a new analogy between ergodic theory and complex algebraic geometry

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.