  Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia
iteration theory of rational functions extend to quasiregular self-maps of the
Riemann sphere for which the degree exceeds the dilatation. We show that in
this context, in contrast to the case of rational functions, the Julia set may
have Hausdorff dimension zero. On the other hand, we exhibit a gauge function
depending on the degree and the dilatation such that the Hausdorff measure with
respect to this gauge function is always positive, but may be finite.
