On the multifractal spectrum of weighted Birkhoff averages

TL;DR

This paper investigates the multifractal spectrum of weighted Birkhoff averages in dynamical systems, establishing continuity, concavity, and explicit spectrum determination for typical weights and specific shift systems.

Contribution

It provides new results on the spectrum's properties and explicit calculations for certain weights and shift systems, extending previous multifractal analysis.

Findings

Spectrum is continuous and concave for uniformly continuous potentials.

Explicit spectrum determination for typical weights with respect to ergodic measures.

Applicability to regular weights like the Möbius sequence in full shift systems.

Abstract

In this paper, we study the topological spectrum of weighted Birkhoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like M\"obius sequence.