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Theory of scale-similar intermittent measures. (English) Zbl 0893.76034

Summary: It is shown that the concept of scale-similar intermittent measures introduced by E. A. Novikov [J. Appl. Math. Mech. 35, 231-241 (1971); translation from Prikl. Mat. Mekh. 35, 266-277 (1971; Zbl 0263.76043)] can be not only the same as that of the stochastic multifractal measures argued by B. B. Mandelbrot [Fractals; Form, chance and dimension. San Fransisco, Freeman (1977; Zbl 0376.28020)] in the context of negative fractal dimensions, but also the same as that of the vast class of general stochastic multifractal measures recently introduced by H. G. E. Hentschel [Phys. Rev. E 50, 243 ff (1994)] – but only if a substantial condition is added to the process of ensemble averaging. The intrinsic probability characterizing the distribution of such a measure is formulated in a general manner so as to be uniquely related to the so-called singularity spectrum \(f(\alpha)\), the intermittency exponents \(\mu(q)\) and the generalized dimensions \(D(q)\). We demonstrate the transformation rule of multifractals, the spatial correlations with any power of such a measure and the special utility of generalized Cantor sets as multifractals. Finally, the multifractal nature of dissipation measure in isotropic turbulence in the inertial range, in which scale-similarity is expected, is discussed in terms of these generalized Cantor sets.

MSC:

76F99 Turbulence
28A80 Fractals
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