  In 2006, A. Gorodetski proved that central fibers of perturbed skew products
are Holder continuous with respect to the base point. In the present paper we
give an explicit estimate of the Holder exponent mentioned above. Moreover, we
extend the Gorodetski theorem from the case when the fiber maps are close to
the identity to a much wider class that satisfy the so-called modified
dominated splitting condition. In many cases (for example, in the case of skew
products over the solenoid or over linear Anosov diffeomorphisms of a torus),
the Holder exponent is close to 1. This allows us in a sense to overcome the
so-called Fubini nightmare. Namely, we prove that the union of central fibers
that are strongly atypical from the point of view of the ergodic theory, has
Lebesgue measure zero, despite the lack of absolute continuity of the holonomy
map for the central foliation. For that we revisit the Hirsch-Pugh-Shub theory,
and estimate the contraction constant of the graph transform map.
