  Generalized derivatives and infinitesimal spaces generalize the idea of
derivatives to mappings which need not be differentiable. It is particularly
powerful in the context of quasiregular mappings, where normal family arguments
imply generalized derivatives always exist. The main result of this paper is to
show that if $f$ is any uniformly quasiregular mapping with $x_0$ a
topologically attracting or repelling fixed point, at which $f$ is locally
injective, then $f$ may be conjugated to a uniformly quasiregular mapping $g$
with fixed point $0$ and so that the infinitesimal space of $g$ at $0$ contains
uncountably many elements. This should be contrasted with the fact that $f$
(and also $g$) is conjugate to $x\mapsto x/2$ or $x\mapsto 2x$ in the
attracting or repelling cases respectively.
