On Regular H\'enon-like Renormalization

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.