ON A GENERAL CONCEPT OF MULTIFRACTALITY - MULTIFRACTAL SPECTRA FOR DIMENSIONS, ENTROPIES, AND LYAPUNOV EXPONENTS - MULTIFRACTAL RIGIDITY

We introduce the mathematical concept of multifractality and describe various multifractal spectra for dynamical systems, including spectra for dimensions and spectra for entropies. We support the study by prov iding some physical motivation and describing several nontrivial examp les. Among them are subshifts of finite type and one-dimensional Marko v maps. An essential part of the article is devoted to the concept of multifractal rigidity. In particular, we use the multifractal spectra to obtain a ''physical'' classification of dynamical systems. For a cl ass of Markov maps, we show that, if the multifractal spectra for dime nsions of two maps coincide, then the maps are differentiably equivale nt. (C) 1997 American Institute of Physics.