- 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
*C*^{2}(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*R*^{2}can be any real number between 0 and*log(d)*. Smillie 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
*C*^{2}. This is a compact, complex manifold*X*which contains*C*^{2}as a dense open subset. Two "obvious" choices are complex projective space*P*^{2}and the product*P*^{1}*x**P*^{1}. A complex Hénon map*f*extends as a birational map*f*_{X}of*X*, and we could study the dynamics of*f*_{X}on*X*. It turns out that for Hénon maps the extension*P*^{2}works well but the extension to*P*^{1}*x**P*^{1}does not. See Section 5 of this expository paper for a discussion of Hénon maps as birational maps. -
Passing to
*P*^{2}adds a 2-sphere*P*^{1}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
*C*^{2}.

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