  This paper develops a conformal renormalization scheme for compact sets $K
\subset \mathbb{C}$. As one application of the conformal renormalization scheme
we prove that for every isolated non-trivial connected component $E \subset K$
there exists a conformal homeomorphism $\phi$ mapping a neighbourhood of $E$
into $\mathbb{C}$ such that the equilibrium measure on $K$ restricted to $E$
equals the scaled push-forward by $\phi^{-1}$ of the equilibrium measure on
$\phi(E)$. Moreover the proof shows that the condition of connectedness of $E$
can be relaxed considerably. We also introduce an inverse to the procedure of
conformal renormalization, which allows one to reconstruct $K$ from its
conformal renormalizations.
