corollary, we prove that for a set $A\subset S^1$ of points with “strongly nonergodic orbits” under the duplication of the circle, the inverse image $q^{-1}(A)$ has Lebesgue measure $0$ (Theorem \[thm:fubini\] below). This is proved despite the fact that the foliation by the fibers $q^{-1}(y), y \in S^1$ may not have an absolutely continuous holonomy.\
The entire Section \[sec:intro\] is concerned with statements of results. In Subsection \[sub:perhol\] below, we introduce our Main Theorem \[thm:mt\] for perturbations of skew products over arbitrary hyperbolic maps. In the next Subsection \[sub:st\] we introduce Theorem \[thm:st\], which improves the Main Theorem in some cases (an important example of which being the solenoid). Two applications of Theorem \[thm:st\] are presented in Subsections \[sub:fubini\] and \[sub:kan\].\
The sections after that will be concerned with proofs. In Sections \[sec:grtrans\] and \[sec:hold\] we work with laminations and graph transform operators, culminating with the proof of the Main Theorem \[thm:mt\]. Several of the results we obtain help us in Section \[sec:st\], where we prove Theorem \[thm:st\]. In Section \[sec:fubini\], we *regain the Fubini property of our central leaves*, thus proving Theorem \[thm:fubini\]. Finally, Section \[sec:app\] consists of an appendix where several technical results are proved.

Persistence and property 