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 not hyperbolically bounded if and only if there is a subspace of $A^{**}$ "almost" isometric to ${ell}_{infty}$ on which ${C_{phi}}^{**}$ "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].