Multifractal analysis for historic set in topological dynamical systems

TL;DR

This paper applies multifractal analysis and topological pressure to historic sets in topological dynamical systems, advancing the understanding of their size and dimension properties, and confirming a conjecture by Olsen.

Contribution

It introduces a novel approach using topological pressure to analyze the multifractal spectrum of historic sets, providing new insights and confirming a prior conjecture.

Findings

Topological pressure effectively describes the size of level sets in historic sets.

The multifractal spectrum of ergodic averages is characterized using topological pressure.

The paper confirms Olsen's conjecture on the multifractal spectrum in this context.

Abstract

In this article, the historic set is divided into different level sets and we use topological pressure to describe the size of these level sets. We give an application of these results to dimension theory. Especially, we use topological pressure to describe the relative multifractal spectrum of ergodic averages and give a positive answer to the conjecture posed by L. Olsen (J. Math. Pures Appl. {\bf 82} (2003)).