A Priori Bounds for H\'enon-like Renormalization

TL;DR

This paper establishes uniform bounds for the renormalization process of Hénon-like maps, providing control over their small-scale geometry and ensuring the pre-compactness of the renormalization sequence, which is crucial for understanding their dynamics.

Contribution

It introduces and proves  priori bounds for He9non-like maps, a key step for analyzing their renormalization and dynamic properties.

Findings

Provides uniform control on small-scale geometry of He9non-like maps.

Ensures pre-compactness of the renormalization sequence.

Lays groundwork for proving renormalization convergence and regularity conditions.

Abstract

We formulate and prove $\textit{a priori}$ bounds for the renormalization of H\'enon-like maps (under certain regularity assumptions). This provides a certain uniform control on the small-scale geometry of the dynamics, and ensures pre-compactness of the renormalization sequence. In a sequel to this paper, a priori bounds are used in the proof of the main results, including renormalization convergence, finite-time checkability of the required regularity conditions and regular unicriticality of the dynamics.