On the multifractal spectrum of weighted Birkhoff averages

Balázs Bárány; Michaƚ Rams; Ruxi Shi

doi:10.3934/dcds.2021199

Article Contents

2022, Volume 42, Issue 5: 2461-2497. Doi: 10.3934/dcds.2021199

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On the multifractal spectrum of weighted Birkhoff averages

1.
Budapest University of Technology and Economics, Department of Stochastics, MTA-BME Stochastics Research Group, P.O.Box 91, 1521 Budapest, Hungary
2.
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
3.
Sorbonne Université, LPSM, 75005 Paris, France

^*Corresponding author: Balázs Bárány
^*Corresponding author: Balázs Bárány

Received: August 31, 2020

Revised: August 31, 2021

Published: May 2022

Balázs Bárány acknowledges support from grants OTKA K123782 and OTKA FK134251. Michaƚ Rams was supported by National Science Centre grant 2019/33/B/ST1/00275 (Poland)

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Abstract

In this paper, we study the topological spectrum of weighted Birk–hoff 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 Möbius sequence.

Keywords:

weighted Birkhoff averages.

Mathematics Subject Classification: Primary: 37C45; Secondary: 37B10, 37B40, 37D35.

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