Renormalization, equipotential annuli and the Hausdorff measure

For a complex single variable polynomial $f$ of degree $d$, let $K$ be its filled Julia set, i.e., the union of all bounded orbits. Assume that $K$ has an invariant component $K^*$ on which $f$ acts as a degree $d_*<d$ map. This is a simplest instance of holomorphic polynomial-like renormalization (