Localized Birkhoff average in beta dynamical systems

Bo Tan; Bao-Wei Wang; Jun Wu; Jian Xu

doi:10.3934/dcds.2013.33.2547

Article Contents

2013, Volume 33, Issue 6: 2547-2564. Doi: 10.3934/dcds.2013.33.2547

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Localized Birkhoff average in beta dynamical systems

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan

Received: January 2012

Revised: October 2012

Published: June 2013

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Abstract

In this note, we investigate the localized multifractal spectrum of Birkhoff average in the beta-dynamical system $([0,1], T_{\beta})$ for general $\beta>1$, namely the dimension of the following level sets $ {x\in [0,1]: \lim_{n\to \infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(T^jx)=f(x)\Big\}, $ where $f$ and $\psi$ are two continuous functions defined on the unit interval $[0,1]$. Instead of a constant function in the classical multifractal cases, the function $f$ here varies with $x$. The method adopted in the proof indicates that the multifractal analysis of Birkhoff average in a general $\beta$-dynamical system can be achieved by approximating the system by its subsystems.

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Mathematics Subject Classification: Primary: 11K55; Secondary: 28A80.

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