  We investigate the rate of convergence of the iterates of an n-dimensional
quasiregular mapping within the basin of attraction of a fixed point of high
local index. A key tool is a refinement of a result that gives bounds on the
distortion of the image of a small spherical shell. This result also has
applications to the rate of growth of quasiregular mappings of polynomial type,
and to the rate at which the iterates of such maps can escape to infinity.
