Historical readings
Here are some papers that show the development of the study of Hénon maps.
 First is Hubbard's paper [H] on Hénon maps.
You can see that he defines the rate of escape function G^{+}.
He is primarily interested in the set U^{+}= {G^{+} >0} and the holomorphic foliation
defined by G^{+}.
A point of view that was adoped later was to observe that G^{+} is the Green function
of K^{+}. From the point of view pluripotential theory, the interesting object to look at
is dd^{c} G^{+}. The operators dd^{c} and
(dd^{c})^{2}, which are the basis of pluripotential theory, are something I have
known and loved since the time I was a student of BA Taylor, and Sibony was equally expert on the subject.
So it is no surprise that both [BS1]
and [FS]
are written with much attention given to the current dd^{c} G^{+}
and the measure (dd^{c}max(G^{+},G^{}))^{2}.

The work of Friedland and Milnor appeared a
couple of years after [H].
It took a different, and very general, point of view. It showed the special role that Hénon maps play inside the group of
all polynomial diffeomorphisms of the plane and introduces the dichotomy between "elementary and affine"
and "Hénon". On one hand, it makes precise the statement: "the elementary and affine maps have
simple dynamics", and
on the other, it explores the dfold horseshoes inside the Hénon family. Read it also as
a model of how to write math.

An early, fascinating phenomenon was the existence of Fatou Bieberbach domains. Such a domain a proper
subset of C^{2} which is biholomorphically
equivalent to C^{2}. These were first obtained as basins of attraction
for automorphisms (cubic Hénon maps). The title of the early paper by Fornaess and Sibony
[FS] emphasizes this
relationship. You should read that paper, and notice the contrast of style and approach when you compare FornaessSibony with
 the early paper of
Hubbard (above),
 the first paper of Hubbard and ObersteVorth, which
has a more topological orientation, including long discussion of various sorts of solenoids; and

[BS1].

The subject of Fatou Bieberbach domains and related issues are taken up by
Rosay and Rudin. They discuss both the original
construction via the uniformization of basins of attraction as well as other, nondynamical, constructions of
Fatou Bieberbach domains.

Michel Hénon was an astronomer. He was interested in numerical study of the Poincaré map
of the real plane to itself,
but it was difficult to numerically simulate the solutions of Newton's equations. So he tried using
a quadratic approximation and found that already these quadratic diffeomorphisms had complicated behavior.
The most famous map was one with a strange attractor.
This is a computer phenomenon: all starting points appear to lead to the same limiting picture, so this
set must be
an attractor, in some sense. However, the behavior of points on the attractor is chaotic, and there seems
to be a positive Lyapunov exponent for points on the attractor.
After you look at the attractor in Hénon's paper, it may be an interesting point of contrast
to look at the
complexification of this map. Find the place on the web page
where it says Henon's famous strange attractor.
Ushiki has drawn the set J inside C^{2}, and you
can see Hénon's attractor inside it.
Since this picture is computed in 4dimensional space
Ushiki's viewer lets you see the projection of the attractor as you rotate it in 4space. I can't describe
it very well, but seeing how it rotates in 4space gives a much clearer picture.

A motivation for the paper [BS1] was the work by Ruelle and Sullivan
[RS]. Ruelle and Sullivan consider diffeomorphisms which
are Axiom A (which means that they are hyperbolic on the nonwandering set, and the periodic points
are dense in the nonwandering set). A hyperbolic map has laminations by stable manifolds and unstable manifolds.
Ruelle and Sullivan define geometric currents, which is similar to being measured laminations. They
show that for Axiom A diffeomorphisms, the stable and
unstable laminations can be given the
structure of geometric currents. This uses the whole theory of Axiom A maps.
On the other hand, any complex Hénon map (hyperbolic or not) has the function G^{+},
and so there are always
currents dd^{c}G^{+} and dd^{c}G^{}. In [BS1] it is shown that if J is a hyperbolic set for
f, then f is Axiom A, and in addition, the currents dd^{c}G^{+} and
dd^{c}G^{} have the structure of geometric currents.
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