A Priori Bounds for Hénon-like Renormalization

We formulate and prove $\mathit{\text{a}}\mathit{\text{priori}}$ 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.