itself that keep the boundary invariant.

*Intermingled attracting basins* means the following thing: the Milnor attractor of the maps mentioned in Theorem \[thm:kan\] consists of the two boundary circles, each one having an attracting basin which is dense and of positive Lebesgue measure.\
In [@IKS08], [@KS*] the theorem above is improved by:\
*The complement to the union of the attracting basins in the perturbed Kan example has Hausdorff dimension smaller than 2.*\
Theorem \[thm:kan\], in a slightly different form, was claimed in [@K94]; as far as we know, the first proof was obtained in [@Bo]. The same tools also allow us to construct diffeomorphisms with intermingled attracting basins [@I08]; the phase space in this case is the product of a solid torus and a circle.

Rate of contraction of the graph transform map {#sec:grtrans}
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In this section we prove statement $a)$ of Theorem \[thm:mt\], and establish the rate of contraction of the graph transform map, see Lemma \[lem:1\] below. There are two ways to prove statement $a)$. The first one is to establish partial hyperbolicity of the $\F $, and refer to the theory. This theory implies the semiconjugacy statement $a)$, but gives no estimate of the rate of 