Shannon entropy for harmonic metrics on cyclic Higgs bundles

Let $X$ be a Riemann surface, $K_X$ its canonical bundle, and $T_X=K_X^{-1}$. A diagonal harmonic metric $h=(h_1,\dots,h_r)$ on a cyclic Higgs bundle of rank $r\geq 2$ associated with a holomorphic $r$-differential $q$ induces Hermitian metrics $H_j=h_j^{-1}\otimes h_{j+1}$ on $T_X$ for $j=1,\dots,r