Renormalization of symmetric bimodal maps with low smoothness

TL;DR

This paper investigates the renormalization of symmetric bimodal maps with low smoothness, establishing the existence of fixed points and demonstrating non-rigidity in the renormalization process.

Contribution

It proves the existence of a renormalization fixed point in $C^{1+Lip}$ symmetric bimodal maps and shows the topological entropy is infinite, revealing non-rigidity.

Findings

Existence of a renormalization fixed point in $C^{1+Lip}$ symmetric bimodal maps

Topological entropy of the renormalization operator is infinite

Presence of a continuum of fixed points indicating non-rigidity

Abstract

This paper deals with the renormalization of symmetric bimodal maps with low smoothness. We prove the existence of the renormalization fixed point in the space $C^{1+Lip}$ symmetric bimodal maps. Moreover, we show that the topological entropy of the renormalization operator defined on the space of $C^{ 1+Lip}$ symmetric bimodal maps is infinite. Further we prove the existence of a continuum of fixed points of renormalization. Consequently, this proves the non-rigidity of the renormalization of symmetric bimodal maps.