Topological description of U via a continuous loop

Determine whether the set U = C \ F, where F is the closure of {t ∉ A_R : λ(t) = log^+|t|}, is the complement of a continuous loop in the complex plane intersecting the unit circle at ζ_3 and ζ̄_3, namely the union of the two interior curves conjectured to support the limiting root measure.

Background

After defining the Lyapunov exponent λ(t), the authors introduce F as the closure of points (outside A_R) where λ(t) equals log^+|t|, and set U=C\F. Numerical evidence suggests U corresponds to the region containing most limiting root mass and includes the unit-circle arc A_L.

A precise topological characterization of U as the complement of a continuous loop intersecting S^1 at ζ_3 and ζ̄_3 would align with the conjectured support of the limiting measure and clarify the global geometry underlying root distributions.