A PRIORI BOUNDS FOR HÉNON-LIKE RENORMALIZATION

The paper proves uniform bounded distortion for Hénon-like renormalizations by reducing the 2D dynamics to a controlled 1D mapping scheme via vertical projections near a dynamical critical value and then applying 1D cross-ratio/Koebe distortion estimates; the argument carefully uses regularity and bounded-type combinatorics to ensure disjointness and uniform length bounds. The candidate solution attempts a direct “hyperbolic sum-of-variations” estimate along horizontal leaves, but it incorrectly assumes that F maps genuine horizontal leaves to horizontal leaves and that forward iterates of two points on a horizontal arc remain on a common horizontal leaf. This invalidates the key pullback-length estimate and the geometric-series bound that follows. The paper’s strategy does not suffer from these issues and appears internally consistent with its stated hypotheses.