Renormalization beyond smoothness: Multimodal maps with mixed combinatorial type

This paper investigates the renormalization of multimodal maps whose smoothness is below C2, for mixed combinatorial type (a−b−c−d), where a,b,c,d∈{4,5,6,7}. We begin by constructing a piecewise affine infinitely renormalizable (PAIR) map for such combinatorial structures, which is then extended to a C1+Lip infinitely renormalizable multimodal (IRMM) map. In the symmetric case where a=d and b=c, the resulting map inherits a corresponding symmetry. More generally, different combinatorial choices lead to distinct IRMMs, each exhibiting unique geometric configurations of their invariant Cantor sets. Beyond existence, we demonstrate that the renormalization operator acting on these C1+Lip maps possesses unbounded topological entropy, reflecting the rich dynamical complexity of the system. Furthermore, by introducing perturbations to the scaling data, we establish the existence of a continuum of renormalization fixed points. This structural flexibility implies the non-rigidity of the associated Cantor attractors, as their defining characteristics vary under deformation.