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Non-normal numbers in dynamical systems fulfilling the specification property

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  • In the present paper we want to focus on the dichotomy of the non-normal numbers -- on the one hand they are a set of measure zero and on the other hand they are residual -- for dynamical system fulfilling the specification property. These dynamical systems are motivated by $\beta$-expansions. We consider the limiting frequencies of digits in the words of the languagse arising from these dynamical systems, and show that not only a typical $x$ in the sense of Baire is non-normal, but also its Cesàro variants diverge.
    Mathematics Subject Classification: Primary: 11K16, 37B10; Secondary: 11A63, 54H20.

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