Renormalization in the H\'enon family, II: The heteroclinic web

TL;DR

This paper investigates the structure of highly dissipative Hénon maps with zero entropy, revealing the density of Morse-Smale maps, the invariance of average Jacobian, and the lamination structure of unstable manifolds through heteroclinic web analysis.

Contribution

It establishes the density of Morse-Smale maps, invariance of average Jacobian in renormalizable cases, and characterizes lamination of unstable manifolds via heteroclinic tangencies in the Hénon family.

Findings

Morse-Smale maps are dense in the parameter region.

The average Jacobian is a topological invariant for renormalizable maps.

Unstable manifolds form a lamination if and only if no heteroclinic tangencies exist.

Abstract

We study highly dissipative H\'enon maps $$ F_{c,b}: (x,y) \mapsto (c-x^2-by, x) $$ with zero entropy. They form a region $\Pi$ in the parameter plane bounded on the left by the curve $W$ of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in $\Pi$, but there exist infinitely many different topological types of such maps (even away from $W$). We also prove that in the infinitely renormalizable case, the average Jacobian $b_F$ on the attracting Cantor set $\OO_F$ is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside $\OO_F$ if and only if there are no heteroclinic tangencies.