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

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 $\mathrm{\Pi }$ 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 $\mathrm{\Pi }$, 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 ${\text{OO}}_F$ 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 ${\text{OO}}_F$ if and only if there are no heteroclinic tangencies.