- The standard context for topological dynamics is a continuous map of a compact space.
The filtration property of complex Hénon maps means that the main dynamical objects of
interest are inside K, which is compact. Thus, if we are interested in the behavior on
K, there is no need to pass to a compactification. And in MAT 655, we do not bother
with a compactification.
In order to describe the entropy of the total
map, you can pass to the one-point compactification of C2 (which is topologically
the same as the 4-sphere). This was used
by Friedland and Milnor to show that
the entropy of the restriction of real Hénon maps to R2 can be any real
number between 0 and log(d).
used the one-point compactification
to show that the entropy of the complex Hénon map is always log(d).
Passing to the one-point compactification as above is passing to the minimal compactification.
Another posibility is to consider a complex compactification of C2. This is a compact,
complex manifold X which contains C2 as a dense open subset. Two "obvious"
choices are complex projective space P2 and the product
P1xP1. A complex Hénon map f extends as a birational
map fX of X, and we could study the dynamics of fX
on X. It turns out that for Hénon maps the extension P2 works well
but the extension to P1xP1 does not. See
Section 5 of this expository paper for a
discussion of Hénon maps as birational maps.
Passing to P2 adds a 2-sphere P1 at infinity.
It is also possible to pass to larger compactifications.
Hubbard and Oberste-Vorth
constructed a compactification which adds a 3-sphere at infinity, and the total space is a closed 4-ball.
In this compactificaton, the closure of K+ meets the 3-sphere at infinity
in a real solenoid. Solenoids have made several appearances in the dynamics of complex Hénon maps.
- A much more complicated compactification was later constructed by
Hubbard, Papadopol and Veselov.
If you are of the point of view that "complicated" = "interesting", then this adds a very interesting
new space at infinity. However, they do not show whether new information can be obtained
about the dynamics of the original map on C2.
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