Multiscaling in multifractals

Abstract

Multiscaling is shown to be a consequence of multifractality when a lower cutoff \ensuremath{\varepsilon} is introduced in calculations of correlation functions. After a suitable rescaling, the correlation function data for different values of \ensuremath{\varepsilon} seem to fall onto a single curve. In the multiscaling regime, however, we show that there is not a unique functional form at varying \ensuremath{\varepsilon}, but a spread very close to a single curve. For each \ensuremath{\varepsilon}, this curve can be computed analytically in terms of the f(\ensuremath{\alpha}) spectrum which characterizes the multifractal. Part of this spectrum can thus be obtained by computing only one moment of the weights at \ensuremath{\varepsilon}\ensuremath{\ne}0.