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

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 $A= \left\{(\omega_k)_{1}^{\infty}\in\Sigma: \omega_{k}\omega_{2k}=0, k\geq 1 \right\},$
$A_{\alpha}= \left\{(\omega_k)_{1}^{\infty}\in\Sigma: \lim\limits_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n}\omega_{k}\omega_{2k}=\alpha \right\}.$

### MSC:

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

### Keywords:

multifractal analysis; ergodic averages; Lyapunov exponents
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