Multifractal Rigidity

Abstract

We consider in this chapter the phenomenon of multifractal rigidity. Roughly speaking, it states that if two dynamical systems are topologically equivalent and some of their multifractal spectra coincide, then the original data must be equivalent (in some sense to be made precise). This leads to a “multifractal classification” of hyperbolic maps (either invertible or noninvertible) in terms of the multifractal spectra. Furthermore, the theory of multifractal analysis has a privileged relation with the experimental study of dynamical systems. In particular, the so-called multifractal spectra, that are obtained from the study of the complexity of the level sets, can be determined experimentally with arbitrary precision. On the other hand, we may be able to recover information about a dynamical system from the information contained in its multifractal spectra. Unfortunately, in general, when we use a single spectrum there is no multifractal rigidity even for topological Markov chains on three symbols.