Recently I met with a problem related to [Stone-Cech Compactification theorem](http://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification)
in Furstenberg's famous paper "non-commuting product."


I try my best to understand Stone-Cech compactification theorem by a not obviously-trivial example. However, when I consider the exponential map in complex plane, I fail to imagine the behaviour for exponential map in compactification space. I guess the compactification space does have complex structure. Can we say more about the further knowledge about the compactification space and the exponential maps in it.


Any comments and remarks will be appreciated.



Stone Cech compactification for exponential map
Convergence in the Stone-Cech compactification is equivalent to convergence of values of *any* bounded continuous function, and there is really a lot of them. In particular, geodesics in symmetric spaces do not converge in the S-C compactification. 


However, I don't think that the S-C compactification has any relevance to products of matrices. There are much more geometrical compactifications and boundaries of the associated symmetric spaces.


