Boundary fixed points in rank-zero Hénon components

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $f:\mathbb C^2\to\mathbb C^2$ be a polynomial diffeomorphism, meaning a polynomial automorphism with polynomial inverse, with nonzero constant complex Jacobian determinant $\delta$. Set \[K^+(f)=\{z\in\mathbb C^2:\{f^n(z):n\geq0\}\text{ is bounded}\}.\] Assume $\operatorname{int}K^+(f)\ne\varnothing$, let $U$ be one of its connected components, and let $g:U\to\mathbb C^2$ be a limit of a subsequence of iterates, uniformly on compact subsets. Rank is complex differential rank. The component is wandering if its forward images are pairwise distinct. Volumes mean four-dimensional Euclidean volume in $\mathbb C^2$. For a periodic component replace $f$ by an iterate so that $f(U)=U$.Assume $U$ is invariant after the indicated replacement by an iterate and $g$ has rank zero, so $g$ is constant with value $x_0$. The eigenvalues at $x_0$ are those of the complex derivative $Df(x_0)$ for this invariant-component iterate.

Statement recorded in the supplied source.

If $x_0\in\partial U$, prove that one eigenvalue at the fixed point $x_0$ equals $1$. [S2]

Short English statement

Prove that a boundary limit fixed point of an invariant rank-zero stable component has an eigenvalue equal to one.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300080 (AMR-052-0080). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld (editor) and conference contributors, Conformal Dynamics Problem List, arXiv:math/9201271v1 (1990), John Milnor, §4, Rank Zero Case and preceding periodic-component reduction, printed p. 10. https://arxiv.org/abs/math/9201271v1.