The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes

TL;DR

This paper establishes exponential contraction of the renormalization operator for infinitely renormalizable unimodal maps of any even degree, demonstrating that their dynamics approach a full renormalization horseshoe, using a unified abstract approach.

Contribution

It proves exponential contraction along hybrid classes for higher degree unimodal maps, extending previous results to all degrees with a unified method.

Findings

Exponential contraction of renormalization for all even degrees.

Construction of the full renormalization horseshoe.

Unified approach applicable to all combinatorics and degrees.

Abstract

We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree $d$. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator ("beau bounds"), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need to account for the varied geometric details of the dynamics, which were the typical source of contraction in previous restricted proofs.