  The invariant class under parabolic and near-parabolic renormalizations
constructed by Inou and Shishikura has been proved to be extremely useful in
recent years. It leads to several important progresses on the dynamics of
certain holomorphic maps with critical points of local degree two. In this
paper, we construct a new class consisting of holomorphic maps with critical
points of local degree three which is invariant under parabolic and
near-parabolic renormalizations. As potential applications, some results of
cubic unicritical polynomials can be obtained similarly as the quadratic case.
For example, the existence of cubic unicritical Julia sets with positive area,
the characterizations of the topology and geometry of cubic irrationally
indifferent attractors etc.
