Journal of the
Korean Mathematical Society JKMS

Let $A$ be a uniform algebra, and let $\phi$ be a self-map of the spectrum $M_A$ of $A$ that induces a composition operator $C_\phi$ on $A$. It is shown that the image of $M_A$ under some iterate $\phi^n$ of $\phi$ is hyperbolically bounded if and only if $\phi$ has a finite number of attracting cycles to which the iterates of $\phi$ converge. On the other hand, the image of the spectrum of $A$ under $\phi$ is $\underline{not}$ hyperbolically bounded if and only if there is a subspace of $A^{**}$ ``almost" isometric to $\ell_\infty$ on which $C_\phi^{**}$ is ``almost" an isometry. A corollary of these characterizations is that if $C_\phi$ is weakly compact, and if the spectrum of $A$ is connected, then $\phi$ has a unique fixed point, to which the iterates of $\phi$ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].

Keywords: uniform algebra, composition operator, hyperbolically bounded, interpolating sequence

MSC numbers: Primary 46J10; Secondary 47B38, 47B48