Compactifications

• 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). 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 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 f_X of X, and we could study the dynamics of f_X 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.