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In this chapter we study a generalization of the regular continued fractions given by E.B. Burger et al. in 2008. We consider a family \(\{T_N:N \geq 1 \}\) of interval maps as generalizations of the Gauss transformation. For the continued fraction expansion arising from \(T_N\) , we solve its Gauss-Kuzmin-type problem by applying the theory of random systems with complete connections by M. Iosifescu. Then we give a two-dimensional Gauss-Kuzmin theorem for N-continued fraction expansions. More precisely, we obtain a Gauss-Kuzmin theorem related to the natural extension of the measure-theoretical dynamical system associated to this expansion.

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N-Continued Fractions

  • 章节

  • 发表日期：

    2025年

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  • Gabriela Ileana Sebe,
  • Dan Lascu

摘要

In this chapter we study a generalization of the regular continued fractions given by E.B. Burger et al. in 2008. We consider a family \(\{T_N:N \geq 1 \}\) of interval maps as generalizations of the Gauss transformation. For the continued fraction expansion arising from \(T_N\) , we solve its Gauss-Kuzmin-type problem by applying the theory of random systems with complete connections by M. Iosifescu. Then we give a two-dimensional Gauss-Kuzmin theorem for N-continued fraction expansions. More precisely, we obtain a Gauss-Kuzmin theorem related to the natural extension of the measure-theoretical dynamical system associated to this expansion.