Zero Lebesgue measure of the λ(t)=0 set in the unit disk

Ascertain whether the set {t ∈ D : λ(t)=0}, where λ(t) is the Lyapunov exponent of the Burau random walk, has Lebesgue measure zero; if so, deduce that the equidistribution of empirical root measures toward ν_bif holds on the entire complex plane.

Background

The Lyapunov exponent λ(t) governs growth of Burau products and the candidate bifurcation measure ν_bif. The authors prove equidistribution toward ν_bif on U=C\F and show that, if the zero set {t:λ(t)=0} has measure zero, the convergence extends to all of C.

Establishing the measure-zero property would remove a key obstruction to global equidistribution and strengthen the connection between random matrix dynamics and Alexander polynomial root distributions.