On compositional dynamics on hardy space

In this work, we examine super-recurrence and super-rigidity of composition operators acting on $H(\mathbb{D})$ the space of holomorphic functions on the unit disk $\mathbb{D}$ and on $H^2(\mathbb{D})$ the Hardy-Hilbert space. We characterize the symbols that generate super-recurrent and super-rigid composition operators acting on $H(\mathbb{D})$ and $H^2(\mathbb{D})$.