Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets

Let A be a uniform algebra, and let ? be a self-map of the spectrum MA of A that induces a composition operator C? on A. The object of this paper is to relate the notion of "hyperbolic boundedness" introduced by the authors in 2004 to the essential spectrum of C?. It is shown that the essential spectral radius of C? is strictly less than 1 if and only if the image of MA under some iterate ?n of ? is hyperbolically bounded. The set of composition operators is partitioned into "hyperbolic vicinities" that are clopen with respect to the essential operator norm. This partition is related to the analogous partition with respect to the uniform operator norm. ? 2006 American Mathematical Society.