What applications of Cantor space are there? Are there well-established applications of the Cantor space ($2^\omega$) in computer science, other than those connected with computable real arithmetic?

John Tucker's page Computation on Topological Data Types mentions applications of the general area in signal processing, 3d graphics, and semantics of process algebras: have these applications been fruitful?
It is used heavily in algorithmic information theory. For example, if you want to define random sequence with respect to a particular measure, you should first define this measure on Borel sets of cantor space.

Compactness of Cantor tree is used implicitly or explicitly in compactness arguments in combinatorics (in particular, it is used in the proof of compactness theorem for boolean functions).