  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 $C^*$--dynamical system ofthe `flip-flop'
automorphism of $\mathcal{O}_2$.
