On Regular H\'enon-like Renormalization

TL;DR

This paper develops a renormalization theory for dissipative Hénon-like maps with bounded combinatorics, incorporating Pesin theory to control dynamics and prove convergence to a universal attractor, extending 1D unimodal results.

Contribution

It introduces a novel renormalization approach for Hénon-like maps using Pesin theory, establishing convergence and regularity properties in higher dimensions.

Findings

Hénon-like maps converge to a universal renormalization attractor.

Renormalization convergence criteria are finite-time checkable.

Infinitely renormalizable maps are regularly unicritical with unique tangency orbit.

Abstract

We develop a renormalization theory of non-perturbative dissipative H\'enon-like maps with combinatorics of bounded type. The main novelty of our approach is the incorporation of Pesin theoretic ideas to the renormalization method, which enables us to control the small-scale geometry of dynamics in the higher-dimensional setting. In a prequel to this paper, it is shown that, under certain regularity conditions on the return maps, renormalizations of H\'enon-like maps have $\textit{a priori}$ bounds. The current paper is devoted to the applications of this critical estimate. First, we prove that H\'enon-like maps converge under renormalization to the same renormalization attractor as for 1D unimodal maps. Second, we show that the necessary and sufficient conditions for renormalization convergence are finite-time checkable. Lastly, we show that every infinitely renormalizable H\'enon-like map is $\textit{regularly unicritical}$: there exists a unique orbit of tangencies between strong-stable and center manifolds, and outside a slow-exponentially shrinking neighborhood of this orbit, the dynamics behaves as a uniformly partially hyperbolic system.