  Theorems and explicit examples are used to show how transformations between
self-similar sets (general sense) may be continuous almost everywhere with
respect to stationary measures on the sets and may be used to carry well known
flows and spectral analysis over from familiar settings to new ones. The focus
of this work is on a number of surprising applications including (i) what we
call fractal Fourier analysis, in which the graphs of the basis functions are
Cantor sets, being discontinuous at a countable dense set of points, yet have
very good approximation properties; (ii) Lebesgue measure-preserving flows, on
polygonal laminas, whose wave-fronts are fractals. The key idea is to exploit
fractal transformations to provide unitary transformations between Hilbert
spaces defined on attractors of iterated function systems. Some of the examples
relate to work of Oxtoby and Ulam concerning ergodic flows on regions bounded
by polygons.
