Shift spaces and attractors in noninvertible horseshoes

As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square I2 in R2 (or more generally, of the cube I in R) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of I2 (or I). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given. 1. Definitions and results. For an integer θ ≥ 2 the setΣθ of all doubly infinite sequences i = (. . . , i−1, i0, i1, . . .), where il ∈ {1, . . . , θ}, equipped with the metric d((. . . , i−1, i0, i1, . . .), (. . . , j−1, j0, j1, . . .)) = ∞ ∑ l=−∞ 2−|l||il − jl| is a Cantor set. The shift mapping σ : Σθ → Σθ given by σ(. . . , i−1, i0, i1, . . .) = (. . . , j−1, j0, j1, . . .) with jl = il+1 is a homeomorphism which defines a simple but nevertheless nontrivial dynamics on Σθ; e.g. its periodic points are dense, and there are dense orbits. Therefore, to ask whether or not a given discrete dynamical system contains a subsystem conjugate to a shift space of this kind is a natural question. Let R be a topological space with metric d, R∗ a compact subset of R and f : R∗ → R continuous. For k ≥ 1 we define the compact sets R∗ k = {p ∈ R | f(p) is defined}, Ak = fk(R∗ k). Then R∗ 1 = R ∗ ⊃ R∗ 2 ⊃ R∗ 3 ⊃ . . . , A1 ⊃ A2 ⊃ . . . , and we consider the compact sets