Narrow Results

We consider a minimal, free action, φ, of the group ℤ^d on the Cantor set X, for d ≥ 1. We introduce the notion of small positive cocycles for such an action. We show that the existence of such cocycles allows the construction of finite Kakutani–Rohlin approximations to the action. In the case, d = 1, small positive cocycles always exist and the approximations provide the basis for the Bratteli–Vershik model for a minimal homeomorphism of X. Finally, we consider two classes of examples when d = 2 and show that such cocycles exist in both.

The Bourrée Part I from Johann Sebastian Bach's Cello Suite No. 3 provides a clear example of structural scaling. The recursive form of this structure can be visualized in the manner of a well known fractal construction — the Cantor set.

Claims emphasizing the supposed freedom of mathematics (e.g. those by Cantor and Hilbert) are considered. A challenge by Frege to such claims is also analyzed and sustained.