The full renormalization horseshoe for multimodal maps

TL;DR

This paper establishes the existence of a full renormalization horseshoe for multimodal maps by analyzing the properties of the renormalization operator, demonstrating its self-homeomorphism on invariant sets.

Contribution

It proves the renormalization operator is a self-homeomorphism on invariant sets and constructs the full renormalization horseshoe for multimodal maps.

Findings

Renormalization operator is a self-homeomorphism on invariant sets.

Existence of the full renormalization horseshoe for multimodal maps.

Provides a new dynamical structure for multimodal maps.

Abstract

In this paper, we consider the renormalization operator $\mathcal R$ for multimodal maps. We prove the renormalization operator $\mathcal R$ is a self-homeomorphism on any totally $\mathcal R$-invariant set. As a corollary, we prove the existence of the full renormalization horseshoe for multimodal maps.