Integral iterations for harmonic maps

We study minimal harmonic maps $g: {$\mathbb{C}$} \to SO(3) \backslash SL(3,{$\mathbb{R}$})$,parameterizedbypolynomialcubicdifferentials$P$ in the plane. The asymptotic structure of such a $g$ is determined by a convex polygon $Y(P)$ in ${\mathbb{R}\mathbb{P}}^2$. We give a conjectural method for determining $Y(P)$ by solving a fixed-point problem for a certain integral operator. The technology of spectral networks and BPS state counts is a key input to the formulation of this fixed-point problem. We work out two families of examples in detail.