Dynamical properties for composition operators on $H^2({\mathbb{C}}_{+})$

Expansivity, Li-Yorke chaos and shadowing are popular and well-studied notions of dynamical systems. Several simple and useful characterizations of these notions within the setting of linear dynamics were obtained recently. We explore these three dynamical properties for composition operators $C_{\phi }f=f\circ \phi$ induced by affine self-maps $\phi$ of the right half-plane ${\mathbb{C}}_{+}$ on the Hardy-Hilbert space $H^2({\mathbb{C}}_{+})$.