The full renormalization horseshoe 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.