The geometry of fractal sets
Two-scale difference equations I: existence and global regularity of solutions
SIAM Journal on Mathematical Analysis
Probability
Two-scale difference equations II. local regularity, infinite products of matrices and fractals
SIAM Journal on Mathematical Analysis
Chaos and Fractals
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It had been suggested in [21] that the existence of mass distributions corresponding to singular continuous measures may imply that "Microphysics is Incomplete". In this paper we point out the mathematical models of time-independent singular continua that are already known, and review the tools available to study them. Fractals constitute one such class of objects. Now, everywhere-continuous but nowhere-differentiable functions can hardly be studied by infinitesimal methods. If, however, they are self-similar under discrete scale transformations, then they may be studied via functional equations with rescaling, which capture precisely this property. Another class of singular continua is provided by random variables which have singular continuous measures as probability distributions, and also satisfy functional equations with rescaling. We conclude that the results so far establish the incompleteness of microphysics only in the time-independent case. In the general case, they suggest problems which appear to be of independent interest as well.