Quantifying the angular parameter linking successive iterates

Characterize the angular parameter θ(t) defined by $(Fn)(t) = Fn−1(θ(t)) in the rear-track iteration, and derive its asymptotic behavior relative to t + 2π; specifically, quantify θ(t) beyond the observed limit θ(t) − (t + 2π) → 0 as t → ∞.

Background

In analyzing convergence and error relations between successive rear-track iterates Fn and Fn−1, the authors introduce θ(t) as the angle on Fn−1 corresponding to the front point of Fn at parameter t.

They can heuristically argue that θ(t) approximates t + 2π for large t, but explicitly state they have no way to quantify θ(t), leaving a precise asymptotic characterization unresolved.