For a complex single variable polynomial $f$ of degree $d$, let $K(f)$ be its filled Julia set, i.e., the union of all bounded orbits. Assume that $K(f)$ 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 (Douady-Hubbard): the dynamics of a higher degree (degree $d$) polynomial $f$ near $K^*$ can be understood in terms of a suitable lower degree (degree $d^{*}$) polynomial to which the restriction $f{|}_{K^{*}}$ is conjugate. One can associate a certain Cantor-like subset $G’$ of the circle with $K^{*}$; the latter is defined in a combinatorial way. We will describe a role the Hausdorff dimension of $G’$ and the respective Hausdorff measure play in geometry of $K^{*}$.