Asymptotic ratio of harmonic measures of sides of a boundary slit

Let l be a Jordan arc in the upper half-plane ℍ with the initial point at z = 0. For ζ ∈ l, let ω⁺(a, ζ) and ω⁻(a, ζ) denote the harmonic measures of the left and right shores of the slit l_ζ⊂ ℍ along the portion of the arc l travelled from 0 to ζ. In one of the seminars within the “Complex Analysis and Integrable Systems” semester held at the Mittag-Leffler Institute in 2011, D. Prokhorov suggested a study of the limit behavior of the quotient ω⁻(a, ζ)/ω⁺(a, ζ) as ζ → 0 along l. In recent publications, D. Prokhorov and his coauthors discussed this problem and proved several results for smooth slits. In this paper, we study the limit behavior of ω⁺(a, ζ)/ω⁻(a, ζ) for a broader class of continuous slits which includes radially and angularly oscillating slits. Some related questions concerning behavior of the driving term of the corresponding chordal Löwner equation are also discussed.