perturbations of over hyperbolic maps in the base. Under the so-called dominated splitting condition, these perturbations have an invariant center lamination (see [@HPS]). It was proven by Gorodetski that this lamination will be continuous, [@G06] (see also [@N] for an earlier particular result). In this paper we estimate the exponent, and prove that in some cases it can be made arbitrarily close to $1$ by making the perturbation small enough.\
Such a center lamination allows us to conjugate any perturbation of a skew product with another skew product, which can be very useful in applications. A priori, the conjugation is only a homeomorphism, and as such it might not behave nicely with the measure. However, the property which we prove gives us some control over this conjugation, and allows to resolve a number of measure-related issues.\
For example, the property allows us to overcome in some sense the Fubini nightmare. In more detail, we prove that perturbations of a skew product over the Smale-Williams solenoid are semiconjugated with the duplication of the circle. We prove that the fibers of this semiconjugacy $q$ (i.e. the manifolds $q^{-1}(y), \ y \in S^1$) are in $y$ with exponent close to $1$. As a 