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.
Highlights
Let (X, T, B, μ) be a measure-theoretic dynamical system.Birkhoff’s ergodic Thm (I)
J=0 where I is the σ-algebra of T -invariant sets. (II)
Consider the size of the set x : lim 1 n−1 f (T jx) = ψ(x) n→∞ n j=0. This is called as localized Birkhoff average : instead of a constant, the function ψ here varies with x
Summary
Let (X, T, B, μ) be a measure-theoretic dynamical system. μ − a.s. Let (X, T, B, μ) be a measure-theoretic dynamical system. J=0 where I is the σ-algebra of T -invariant sets. T is ergodic, lim 1 n−1 f (T jx) = n→∞ n j=0 f (x)dμ μ − a.s. Multifractal analysis of Birkhoff average : Classical case :. Remark : The Birkhoff average of a point x should depend on x itself. This is called as localized Birkhoff average : instead of a constant, the function ψ here varies with x
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