Multifractal analysis of weighted ergodic averages

TL;DR

This paper investigates the multifractal properties of weighted ergodic averages within symbolic dynamics, introducing a thermodynamical formalism applicable to various weights, and discusses open problems for irregular weights.

Contribution

It develops a thermodynamical formalism for multifractal analysis of weighted ergodic averages in symbolic dynamics, extending to different types of weights.

Findings

Thermodynamical formalism applies to stationary ergodic and uniquely ergodic weights.

The formalism's applicability to irregular weights like the Möbius function remains unresolved.

The paper outlines open problems in the field.

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 thermodynamical formalism which leads to a multifractal spectrum. It is proved that this thermodynamical formalism applies to different kinds of dynamically defined weights, including stationary ergodic random weights, uniquely ergodic weights etc. But the validity of the thermodynamical formalism for very irregular weights, like M\"{o}bius function, is an unsolved problem. The paper ends with some other unsolved problems.