Multifractality of self-affine fractals

The concept of multifractality is extended to self-affine fractals in order to provide a more complete description of fractal surfaces. We show that for a class of iteratively constructed self-affine functions there exists an infinite hierarchy of exponents Hₐ describing the scaling of the qth order height-height correlation function cₐ (x) xₐ^qH. Possible applications to random walks and turbulent flows are discussed. It is demonstrated on the example of random walks along a chain that for stochastic lattice models leading to self-affine fractals Hₐ exhibits phase-transition-like behavior.