Cowen-Douglas function and its application on chaos

In this paper, on $\mathbb{D}$ we define Cowen-Douglas function introduced by Cowen-Douglas operator $M_{\varphi }^{\ast }$ on Hardy space ${\mathcal{H}}^2(\mathbb{D})$ and we give a sufficient condition for Cowen-Douglas function, where $\varphi \in {\mathcal{H}}^{\mathrm{\infty }}(\mathbb{D})$. Moreover, we give some applications of Cowen-Douglas function on chaos, such as application on the inverse chaos problem for $\varphi (T)$, where $\varphi$ is a Cowen-Douglas function and $T$ is the backward shift operator on ${\mathcal{L}}^2(\mathbb{N})$.