On the Thin Boundary of the Fat Attractor

For, \(0<\lambda <1\), consider the transformation \(T(x) = d x \) (mod 1) on the circle \(S^1\), a \(C^1\) function \(A:S^1 \rightarrow \mathbb {R}\), and, the map \(F(x,s) = ( T(x) , \lambda \, s + A(x))\), \((x,s)\in S^1 \times \mathbb {R}\). We denote \(\mathscr {B}= \mathscr {B}_\lambda \) the upper boundary of the attractor (known as fat attractor). We are interested in the regularity of \(\mathscr {B}_\lambda \), and, also in what happens in the limit when \(\lambda \rightarrow 1\). We also address the analysis of the following conjecture which were proposed by R. Bamón, J. Kiwi, J. Rivera-Letelier and R. Urzúa: for any fixed \(\lambda \), \(C^1\) generically on the potential A, the upper boundary \(\mathscr {B}_\lambda \) 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.