base 2. We say that $y$ is $\kappa,w-$*atypical* if $\om^+$ is $\kappa
,w-$atypical. Let $A_{\kappa,w}\subset S^1$ be the set of atypical points. By the ergodic theorem, it has Lebesgue measure 0 in $S^1$.

\[thm:fubini\] Consider a $\F $ over the solenoid map as in Theorem \[thm:st\]. Let $\zeta : Z \to Z$ be the duplication of a circle. For any word $w$ and positive $\ka $ $\rho $ the following holds. Let $A_{\ka ,w}$ be the same as above. Then for any $\rho $ perturbation $\G $ of $\F $ and $q$ as in , we have: $$\label{eqn:semifub2} \emph{mes }q^{-1}(A_{\kappa,w}) = 0.$$ In other words, the union of $\kappa,w-$atypical fibers has Lebesgue measure 0 in $X$.

An analog of this theorem for perturbations of skew products over the Anosov of a two-torus may be proved basing on a recent result of P. Saltykov [@S09].

Attractors with intermingled basins {#sub:kan}
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The tools developed in this paper allow us to get a new proof of the following phenomenon discovered by I. Kan:

\[thm:kan\] \[[@IKS08], [@KS*]\] The set of maps of an annulus $S^1\times [0,1]$ that have intermingled attracting basins is open in the set of all maps of the annulus into 