  It is well-known that a holomorphic function near a repelling fixed point may
be conjugated to a linear function. The function which conjugates is called a
Poincar\'e linearizer and may be extended to a transcendental entire function
in the plane. In this paper, we study the dynamics of a higher dimensional
generalization of Poincar\'e linearizers. These arise by conjugating a
uniformly quasiregular mapping in $\R^m$ near a repelling fixed point to the
mapping $x\mapsto 2x$. In particular, we show that the fast escaping set of
such a linearizer has a spider's web structure.
