Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems

Abstract

Power-law sensitivity to the initial conditions at the edge of chaos provides a natural relation between the scaling properties of the dynamics attractor and its degree of nonextensivity within the generalized statistics recently introduced by one of the authors (C.T.) and characterized by the entropic index $q$. We show that general scaling arguments imply that $1/(1\ensuremath{-}q)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/{\ensuremath{\alpha}}_{\mathrm{min}}\ensuremath{-}1/{\ensuremath{\alpha}}_{\mathrm{max}}$, where ${\ensuremath{\alpha}}_{\mathrm{min}}$ and ${\ensuremath{\alpha}}_{\mathrm{max}}$ are the extremes of the multifractal singularity spectrum $f(\ensuremath{\alpha})$ of the attractor. This relation is numerically verified in standard $D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ dissipative maps.