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 $K^{\ast }$ on which $f$ acts as a degree $d_{\ast }<d$ map. This is a simplest instance of holomorphic polynomial-like renormalization (Douady-Hubbard). One can associate a certain Cantor-like subset $G^{\mathrm{\prime }}$ of the circle with $K^{\ast }$; it is defined as the set of arguments of all smooth or broken rays to $K^{\ast }$. We will describe a role the Hausdorff dimension of $G^{\mathrm{\prime }}$ and the respective Hausdorff measure play in geometry of $K^{\ast }$. In particular, we give upper and lower bounds on the modulus of renormalization in terms of the Hausdorff measure of $K^{\ast }$.