Multifractal analysis of some multiple ergodic averages for the systems with non-constant Lyapunov exponents. (English) Zbl 1323.28015

Liao, Lingmin; Rams, Michał

Real Anal. Exch. 39(2013-2014), No. 1, 1-14 (2014).

Let \(\{f_{0}; f_{1}\}\) be the iterated function system on \([0; 1]\), \(f_{0}(x) = e^{-\lambda_{0}}x\), \(f_{1}(x) = e^{-\lambda_1}x + 1 - e^{-\lambda_1}\). Here \(\lambda_{0,1} > 0\) and \(e^{-\lambda_{0}}+e^{-\lambda_1}\leq 1\). It has the usual symbolic description by \(\Sigma = \{0, 1\}^{\mathbb N}\) with a natural projection \[ \pi(\omega)=\lim\limits_{n\to\infty} f_{\omega_1}\circ f_{\omega_2}\circ \dots \circ f_{\omega_n}. \] The authors calculate the Hausdorff dimensions of the sets \(\pi(A)\) and \(\pi(A_{\alpha})\), where \[ A= \left\{(\omega_k)_{1}^{\infty}\in\Sigma: \omega_{k}\omega_{2k}=0, k\geq 1 \right\}, \]
\[ A_{\alpha}= \left\{(\omega_k)_{1}^{\infty}\in\Sigma: \lim\limits_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n}\omega_{k}\omega_{2k}=\alpha \right\}. \]

Reviewer: Boris A. Kats (Kazan)

MSC:

28A80	Fractals
37C45	Dimension theory of smooth dynamical systems
28A78	Hausdorff and packing measures

Keywords:

multifractal analysis; ergodic averages; Lyapunov exponents