A geometric platonic approach to multifractality and turbulence

A large family of deterministic measures built as projections off fractal functions has been introduced recently.^1,2 These constructed measures, which transform either simple multifractals as defined by deterministic cascades or uniform measures via fractal interpolating functions,^4,5 range from nontrivial multifractals to absolutely continuous measures, and include as a limiting case the Gaussian distribution.⁶ In this work, examples of deterministic measures which possess multiscaling properties when analyzed following the multifractal formalism are given. It is shown that suitable measures which not only preserve the singularity spectrum of turbulence but also capture the inherent details present in turbulent data sets may be obtained. These results suggest a plausible deterministic framework for the study of intermittent phenomena via a new geometric approach - one which closely evokes Plato's ideas that all we observe may be "shadows" of simple objects.⁷.