Fractional integral associated to the self-similar set or the generalized self-similar set and its physical interpretation

This paper is based on a study of Nigmatullin [Teer. Mat. Fit. 90 (1992) 354]. When the ''residual'' memory set is a self-similar set which is generated by similarities S(j)x=xi(j)x+b(j) (0<xi(j) <1, b(2)<...<b(K)=t(1-xi(K)), j=1, 2,..., K) on [0, t] or a generalized self-similar set which is generated by a family of similarities {S-nj(x)=xi(n,j)x+b(n,j):0<xi(n,j)<1, b(n,j)is an element of R, j=1,2,...,K-n}(n is an element of Z+) on [0, t], we prove that the fractional exponent of the fractional integral is not uniquely determined by the fractal dimension of the self-similar set or generalized self-similar set, it is determined by In P-1/In xi(1) of the self-similar measure mu=Sigma(j=1)(K) P-j mu circle S-j(-1), 0<P-j<1, Sigma(j=1)(K)P(j)=1 on this self-similar set or of the generalized self-similar measure mu'=Sigma(j=1)(infinity) P-j mu'circle S-j(-1), 0<P-j<1, Sigma(j=1)(infinity) P-j=1 on the generalized self-similar set, and it can have the value of all positive real numbers. Our results generalize and extend the results of Nigmatullin.