  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.
