Let A be a uniform algebra, and let

be a self-map of the spectrum

of A that induces a composition operator

, on A. It is shown that the image of

under some iterate

of \phi is hyperbolically bounded if and only if \phi has a finite number of attracting cycles to which the iterates of

converge. On the other hand, the image of the spectrum of A under

is not hyperbolically bounded if and only if there is a subspace of

"almost" isometric to

on which

"almost" an isometry. A corollary of these characterizations is that if

is weakly compact, and if the spectrum of A is connected, then

has a unique fixed point, to which the iterates of

converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].