Sharp Attraction Thresholds for Neutral Jordan Skew Products

For a class of holomorphic skew products tangent to the identity, we give necessary and sufficient conditions for an open attracting domain and for an open domain attracted along the base axis. The transverse leading coefficient is a fixed matrix with real spectrum and arbitrary Jordan blocks. If the base is \(g(z)=z-z^{2t+1}+az^{3t+1}+O(z^{3t+2})\) and the fiber multiplier is \(I-iz^tB-cz^{2t}I+R(z)\), with \(R(z)=O(z^{2t+1})\) in the algebra generated by \(B\), the attraction threshold is

\[\operatorname{Re}c>\max_{\beta\in\operatorname{spec}B}\left\{\frac{\beta^2}{2}+\beta\operatorname{Im}a+t(m_\beta-1)\right\}.\]

Attachment to the base axis requires the same strict inequality with one added to its right-hand side. Here \(m_\beta\) is the largest Jordan block size at \(\beta\). We prove exact rates at every Jordan height, including the boundary cases, and construct invariant open domains. The criteria distinguish equal-spectrum examples and exhibit the effect of a higher base coefficient.

These are complete finite-jet criteria within the stated linear-fiber class, not a solution of the general AIM question about degenerate characteristic directions. This AI-assisted preprint is self-audited and unrefereed; no independent review, formal verification or absolute-priority certification is claimed.