Let A be a uniform algebra, and let φ be a self-map of the spectrum MA of A that induces a composition operator Cφ on A. It is shown that the image of MA under some iterate φn of φ is hyperbolically bounded if and only if φ has a finite number of attracting cycles to which the iterates of φ converge. On the other hand, the image of the spectrum of A under φ is not hyperbolically bounded if and only if there is a subspace of A** "almost" isometric to ℓ∞ on which Cφ** is "almost" a,n isometry. A corollary of these characterizations is that if Cφ is weakly compact, and if the spectrum of A is connected, then φ has a unique fixed point, to which the iterates of φ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz .
Galindo, P., Gamelin, T. W., & Lindström, M. (2004). Composition operators on uniform algebras and the pseudohyperbolic metric. Journal of the Korean Mathematical Society, 41(1), 1–20. https://doi.org/10.4134/JKMS.2004.41.1.001