  Renormalizations can be considered as building blocks of complex dynamical
systems. This phenomenon has been widely studied for iterations of polynomials
of one complex variable. Concerning non-polynomial hyperbolic rational maps, a
recent work of Cui-Tan shows that these maps can be decomposed into
postcritically finite renormalization pieces. The main purpose of the present
work is to perform the surgery one step deeper. Based on Thurston's idea of
decompositions along multicurves, we introduce a key notion of Cantor
multicurves (a stable multicurve generating infinitely many homotopic curves
under pullback), and prove that any postcritically finite piece having a Cantor
multicurve can be further decomposed into smaller postcritically finite
renormalization pieces. As a byproduct, we establish the presence of separating
wandering continua in the corresponding Julia sets. Contrary to the polynomial
case, we exploit tools beyond the category of analytic and quasiconformal maps,
such as Rees-Shishikura's semi-conjugacy for topological branched coverings
that are Thurston-equivalent to rational maps.
