ANALYTIC-EXPRESSION FOR THE STRUCTURE FACTOR AND FOR THE MOMENT-GENERATING FUNCTION OF FRACTAL SETS AND MULTIFRACTAL MEASURES

By applying the definition of Markov operator for iterated function sy stems with probability, an analytic expression for the structure facto r and for the moment-generating function of a class of fractal sets is obtained in the presence of arbitrary multifractal distributions asso ciated with affine iterated function systems. The properties of the st ructure factor of fractal sets are analysed in detail. The results dev eloped for the structure factors have interesting implications as rega rds to the theory of Laplace transforms of singular non-atomic measure s. Some physical applications of the theory, related to fractal and he terogeneous systems in the case of adsorption and relaxation, are brie fly addressed. Integral transform theory is also applied to solve the Dirichlet problem for the Laplace equation on the circle in the presen ce of singular non-atomic boundary conditions.