  We investigate the distribution of energy density in a stationary
self-reproducing inflationary universe. We show that the main fraction of
volume of the universe in a state with a given density at any given moment of
proper time t is concentrated near the centers of deep exponentially wide
spherically symmetric wells in the density distribution. Since this statement
is very surprising and counterintuitive, we perform our investigation by three
different analytical methods to verify our conclusions, and then confirm our
analytical results by computer simulations. If one assumes that we are typical
observers living in the universe at a given moment of time, then our results
may imply that we should live near the center of a deep and exponentially large
void, which we will call infloid. Validity of this particular interpretation of
our results is not quite clear since it depends on the as-yet unsolved problem
of measure in quantum cosmology. Therefore at the moment we would prefer to
consider our results simply as a demonstration of nontrivial properties of the
hypersurface of a given time in the fractal self-reproducing universe, without
making any far-reaching conclusions concerning the structure of our own part of
the universe. Still we believe that our results may be of some importance since
they demonstrate that nonperturbative effects in quantum cosmology, at least in
principle, may have significant observational consequences, including an
apparent violation of the Copernican principle.
