Multifractal analysis of weighted ergodic averages

Abstract

We propose to study the multifractal behavior of weighted ergodic averages. Our study in this paper is concentrated on the symbolic dynamics. We introduce a thermodynamic formalism which leads to a multifractal spectrum. It is proved that this thermodynamic formalism applies to different kinds of dynamically defined weights, including stationary ergodic random weights, uniquely ergodic weights etc. But the validity of the thermodynamic formalism for very irregular weights, like Möbius function, is an unsolved problem. The paper ends with some other unsolved problems.