set $Y$ such that the restriction of $\G|Y$ is conjugated to a skew product close to $\F|\L \times M$, in the sense of , and .**

The solenoid case {#sub:st}
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In this section we will present a partial improvement of Theorem \[thm:mt\] that is inspired by the example of the Smale-Williams solenoid. Let us begin by introducing and describing the solenoid map. Fix constants $R\geq 2$ and $\lambda<0.1$, whose particular values will not be important. Let $B$ denote the solid torus $$B = S^1 \times D, \textrm{ where }\ S^1 = \{y \in \rr
/\zz\}, \ D = \{z \in \mathbb C| |z| \le R \}.$$ The *solenoid map* is defined as $$\label{eqn:sol} h:B \rightarrow
B, \textrm{ }\textrm{ }\textrm{ } h(y, z) = (2y, e^{2\pi i y} +
\lambda z).$$ The maximal attractor of the solenoid map: $$\Lambda = \bigcap_{k=0}^\infty h^k(B)$$ is called the *Smale-Williams solenoid*. It is a hyperbolic invariant set with contraction coefficient $\lambda$ and expansion coefficient $\mu^{-1} = 2$ (we take the sup norm in $T_bB$ in the coordinates $y,z$). Moreover, the estimates in hold with $\lambda=\lambda_-$ and $\mu=\mu_-$.\
We can generalize the above to the following setup: let $B=Z\times
F$ be the product of two 