  We fix $SU(2)$ lattice gauge fields to the Maximally Abelian gauge in both
three and four dimensions. We extract the corresponding $U(1)$ fields and
monopole current densities and calculate separately the confining string
tensions arising from these $U(1)$ fields and monopole `condensates'. We
generate multiple Gribov copies and study how the $U(1)$ fields and monopole
distributions vary between these different copies. As expected, we find
substantial variations in the number of monopoles, their locations and in the
values of the $U(1)$ field strengths. The string tensions extracted from
`extreme' Gribov copies also differ but this difference appears to be no more
than about 20\%. We also directly compare the fields of different Gribov
copies. We find that on the distance scales relevant to confinement the $U(1)$
and monopole fluxes that disorder Wilson loops are highly correlated between
these different Gribov copies. All this suggests that while there is indeed a
Gribov copy problem the resulting ambiguity is, in this gauge and for the study
of confinement, of limited importance.
