On the non-Archimedean Hitchin map for SL₂(F)

Let F be a non-Archimedean valued field, $Σ$ a closed Riemann surface of genus at least two, and $Γ$ its fundamental group. Building on the theory of equivariant harmonic maps into R-trees, we study the non-Archimedean Hitchin map from the SL₂(F)-character variety XF(Γ), equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on $Σ$. We prove that this map is continuous and that its image is contained in the space of Jenkins--Strebel differentials. Moreover, we establish a dynamical characterization of unbounded representations, showing that the induced action of $Γ$ on the Bruhat--Tits tree of SL₂(F) is never small.