The relationship between the fractional integral and the fractal structure of a memory set

It is shown that there is no direct relation between the fractional exponent v of the fractional integral and the fractal structure of the memory set considered, v depends only the first contraction coefficient χ1 and the first weight P1 of the self-similar measure (or infinite self-similar measure) μ on the memory set. If and only if P1=χ1β (where β ∈ (0,1) is the fractal dimension of the memory set), v is equal to the fractal dimension of the memory set. It is also true that v is continuous about χ1 and P1.