Multifractality of self-affine fractals

Abstract

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 ${\mathit{H}}_{\mathit{q}}$ describing the scaling of the qth order height-height correlation function ${\mathit{c}}_{\mathit{q}}$(x)∼${\mathit{x}}_{\mathit{q}}^{\mathit{q}\mathit{H}}$. 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 ${\mathit{H}}_{\mathit{q}}$ exhibits phase-transition-like behavior.