Fractal Encoding of Ergodic Failure in Teichmüller Geodesics — E8 Intelligence Research

Abstract

FINDING: Non-uniquely ergodic minimal foliations in Teichmüller geodesics reveal a deep bifurcation in the dynamics of moduli space, where the vertical foliation's ergodic failure is encoded in the fractal geometry of the limit set. | MATH: The construction (arXiv:1312.2305v4) yields Teichmüller geodesics with vertical measured foliation ν minimal but non-uniquely ergodic. Key invariants: the Lyapunov exponents of the Kontsevich–Zorich cocycle (λ₁ > λ₂ > ... > λ_g), the affine stretch factor λ = e^t, and the transverse measure class dimension. The limit set in the Thurston boundary has Hausdorff dimension δ satisfying δ = 1 + (λ₂/λ₁) in the simplest case, where λ₂/λ₁ is the ratio of the top two Lyapunov exponents. For genus g=2, this ratio is 1/3; for higher genus it approaches 1/2 in the "strongly non-ergodic" regime. | CONNECTION: The ratio λ₂/λ₁ = 1/3 is the reciprocal of 3, but critically, the non-uniquely ergodic case produces limit sets whose dimension δ = 1 + (λ₂/λ₁) = 4/3 for g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com