  For, $0<\lambda<1$, consider the transformation $T(x) = d x $ (mod 1) on the
circle $S^1$, a $C^1$ function $A:S^1 \to \mathbb{R}$, and, the map $F(x,s) = (
T(x) , \lambda \, s + A(x))$, $(x,s)\in S^1 \times \mathbb{R}$. We denote
$\mathcal{B}= \mathcal{B}_\lambda$ the upper boundary of the attractor (known
as fat attractor). We are interested in the regularity of
$\mathcal{B}_\lambda$, and, also in what happens in the limit when $\lambda\to
1$. We also address the analysis of the following conjecture which were
proposed by R. Bam\'on, J. Kiwi, J. Rivera-Letelier and R. Urz\'ua: for any
fixed $\lambda$, $C^1$ generically on the potential $A$, the upper boundary
$\mathcal{B}_\lambda$ is formed by a finite number of pieces of smooth unstable
manifolds of periodic orbits for $F$. We show the proof of the conjecture for
the class of $C^2$ potentials $A(x)$ satisfying the twist condition (plus a
combinatorial condition). We do not need the generic hypothesis for this
result. We present explicit examples. On the other hand, when $\lambda$ is
close to $1$ and the potential $A$ is generic a more precise description can be
done. In this case the finite number of pieces of $C^1$ curves on the boundary
have some special properties. Having a finite number of pieces on this boundary
is an important issue in a problem related to semi-classical limits and
micro-support. This was consider in a recent published work by A. Lopes and J.
Mohr. Finally, we present the general analysis of the case where $A$ is
Lipschitz and its relation with Ergodic Transport
