On the Thin Boundary of the Fat Attractor

Abstract

For, 0 < λ < 1, consider the transformation T (x) = dx (mod 1) on the circle S 1 , a C 1 function A : S 1 → R, and, the map F(x, s) = (T (x), λ s + A(x)), (x, s) ∈ S 1 × R. We denote B = B λ the upper boundary of the attractor (known as fat attractor). We are interested in the regularity of B λ , and, also in what happens in the limit when λ → 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 λ , C 1 generically on the potential A, the upper boundary B λ 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 λ 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.

Applications of variable discounting dynamic programming to iterated function systems and related problems

“…In [Tsu01] (see also [BKRLU06] for topological properties of the attractor) the author gives a description of the SRB measure, by analyzing S(x, a) and conjectured that the optimal return function sup a S(x, a) can be used to describe the boundary of the attractor. This conjecture was partially solved in [LO14], assuming that the potential f satisfies a certain twist condition. A natural question arises when we change the skew map F by a non uniform hyperbolic one, with variable discount such as G(x, y) = (T (x), ln(1 + y) + f (x)) ( note that {1, 2} is always contained in the spectrum of DG(x, 0) ).…”

On the connection between a skew product IFS and the ergodic optimization for a finite family of potentials

“…In [Tsu01] this map was studied from a measure theoretical point of view, giving a description of the SRB measure. This kind of skew product and its connection with iterated function systems (IFS) theory received great attention in the last few years (see, for example, [DGR17] for skew products involving diffeomorphisms in manifolds, [Ram03] for non linear Baker maps and [Tsu01], [BKRLU06], [LO18] for the linear ones).…”

“…In [LO18], question (b) was studied and a partial answer to the conjecture on the structure of the boundary of the attractor was given. The authors prove that the superior boundary of the attractor is a piecewise differentiable graph (x, v λ (x)) under some hypothesis.…”

Selection of calibrated subaction when temperature goes to zero in the discounted problem

“…This function is unique and we call b λ the λ-calibrated subaction for A (see for instance Theorem 1 in [4] or [16]). The solution b λ can be obtained in the following way: consider τ j , j = 1, ..., d the inverse branches of T .…”

“…From (6) in [16] we get that b λ (y) = S λ,A (y, a(y)), where a(y) is a realizer of y, then for any y we have that b λ (y) = 1 (1 − λ)…”