Period tripling and quintupling renormalizations below $C^2$ space

TL;DR

This paper investigates the existence and properties of renormalization fixed points for period tripling and quintupling in unimodal maps below the $C^2$ regularity, revealing a rich structure of fixed points and unbounded entropy.

Contribution

It demonstrates the existence of renormalization fixed points on piece-wise affine maps and extends these to $C^{1+Lip}$ unimodal maps, uncovering a continuum of fixed points and unbounded entropy.

Findings

Existence of fixed points for period tripling and quintupling renormalizations.

Extension of fixed points to $C^{1+Lip}$ unimodal maps.

Unbounded topological entropy of the renormalization operators.

Abstract

In this paper, we explore the period tripling and period quintupling renormalizations below $C^2$ class of unimodal maps. We show that for a given proper scaling data there exists a renormalization fixed point on the space of piece-wise affine maps which are infinitely renormalizable. Furthermore, we show that this renormalization fixed point is extended to a $C^{1+Lip}$ unimodal map, considering the period tripling and period quintupling combinatorics. Moreover, we show that there exists a continuum of fixed points of renormalizations by considering a small variation on the scaling data. Finally, this leads to the fact that the tripling and quintupling renormalizations acting on the space of $C^{1+Lip}$ unimodal maps have unbounded topological entropy.