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Theorems and explicit examples are used to show how transformations betweenself-similar sets (general sense) may be continuous almost everywhere withrespect to stationary measures on the sets and may be used to carry well knownflows and spectral analysis over from familiar settings to new ones. The focusof this work is on a number of surprising applications including (i) what wecall fractal Fourier analysis, in which the graphs of the basis functions areCantor sets, being discontinuous at a countable dense set of points, yet havevery good approximation properties; (ii) Lebesgue measure-preserving flows, onpolygonal laminas, whose wave-fronts are fractals. The key idea is to exploitfractal transformations to provide unitary transformations between Hilbertspaces defined on attractors of iterated function systems. Some of the examplesrelate to work of Oxtoby and Ulam concerning ergodic flows on regions boundedby polygons.

Conjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces