Divergence points of deformed empirical measures. (Q1812516)

The author proposes a multi-fractal framework based on deformations of empirical measures. By considering particular deformations, essentially all multi-fractal spectra are obtained. Upper and lower bounds for the multi-fractal spectrum for an arbitrary continuous deformation are given. A variational principle for the general multi-fractal spectrum is provided. It is proved that for a large class of deformations the set of divergence points has full dimension. As an application the author states a variational principle for a very general type of mixed multi-fractal spectrum of ergodic averages.