We formulate and prove a priori bounds for the renormalization of Hénon-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.

Submitted 20 November, 2024; originally announced November 2024.

arXiv:2411.13624