Abstract
For a very special class of symbol sequences, both simple and complicated deterministic chaotic systems universally generate the same multiplicative multifractal generating functions. This new universality is based upon statistical independence that is deterministic in origin and accounts for the evidence that turbulence can mimic certain simple pseudo-random number generators. Our result follows from the construction of classes of symbol sequences that are a generalization of normal numbers and which occur universally for a certain class of dynamical systems. By constructing an example of the required class of symbol sequence, we are able to provide a fully deterministic explanation for thef(α) spectra of the sort obtained for one-dimensional cuts of turbulence. The application to turbulence in three dimensions is also discussed, as are deterministic noise and the application to experimental time series. The universality limits the extent within which one can infer the details of the dynamics from observedf(α) spectra.
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McCauley, J.L. Universal deterministic statistical independence. Z. Physik B - Condensed Matter 81, 115–129 (1990). https://doi.org/10.1007/BF01454223
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DOI: https://doi.org/10.1007/BF01454223