  I apply the newly developed dynamical cluster approximation (DCA) to the
calculation of the electron and phonon dispersions in the two dimensional
Holstein model. In contrast to previous work, the DCA enables the effects of
spatial fluctuations (non-local corrections) to be examined. Approximations
neglecting and incorporating lowest-order vertex corrections are investigated.
I calculate the phonon density of states, the renormalised phonon dispersion,
the electron dispersion and electron spectral functions. I demonstrate how
vertex corrections stabilise the solution, stopping a catastrophic softening of
the $(\pi,\pi)$ phonon mode. A kink in the electron dispersion is found in the
normal state along the $(\zeta,\zeta)$ symmetry direction in both the vertex-
and non-vertex-corrected theories for low phonon frequencies, corresponding
directly to the renormalised phonon frequency at the $(\pi,0)$ point. This kink
is accompanied by a sudden drop in the quasi-particle lifetime. Vertex and
non-local corrections enhance the effects at large bare phonon frequencies.
