Representations of the Cuntz algebra ${\mathcal{O}}_{N}$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized by using the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi, and Lascu. Furthermore, a certain symmetry of such an interval dynamical system is interpreted as a covariant representation of the $C^{\ast }$-dynamical system of the “flip-flop” automorphism of ${\mathcal{O}}_{2}$.

In this paper we determine the ${{L}^{1}}\to {{L}^{1}}$ and ${{L}^{\infty }}\to {{L}^{\infty }}$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

It is proved in this paper that for any non-elementary subgroup $G$ of $\mathrm{PSL}(2,\sGa_n)$, which has no elliptic element, to be not strict, there is a minimal generating system of $G$ consisting of loxodromic elements, and that if $G$ is a non-elementary subgroup of $\mathrm{PSL}(2,\sGa_n)$ of which each loxodromic element is hyperbolic, then $G$ is conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$.

AMS 2000 Mathematics subject classification: Primary 30F40. Secondary 20H10