Mandelbrot Cascade Measures Independent of Branching Parameter

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Abstract

Mandelbrot cascade measures were introduced to explain intermittency in fully developed turbulence. They are defined by the scale hierarchy with a fixed branching parameter c and by the distribution of breakdown coefficients which are responsible for the transport of energy from larger to smaller scales. We show that the measures corresponding to both conservative and nonconservative cascades strongly depend on the parameter c. In particular, only Lebesgue measure can be generated by a cascade process with an arbitrary integer c.

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