Dynamical behaviour of Taylor shifts

Let $\Omega$ be an open subset of the Riemann sphere and let $H(\Omega)$ be the Fréchet space of functions holomorphic in $\Omega$ and vanishing at $\infty$, endowed with the topology of compact convergence. If $0 \in \Omega$, the  Taylor shift $T: H(\Omega) \to H(\Omega)$ is defined by $$(Tf)(z):=\begin{cases}(f(z)-f(0))/z, & z \not =0\\   f'(0), & z=0 \end{cases}.$$ Locally at $0$, the operator $T$ acts as backward shift on the Taylor coefficients. In this talk, results about the topological and metric dynamical behaviour of $(H(\Omega),T)$ and $(A^p(\Omega),T)$, where $A^p(\Omega)$ denotes the Bergman space of $p$-integrable functions in $H(\Omega)$, are presented.