Topological horseshoes for Arneodo–Coullet–Tresser maps

In this paper, we study the family of Arneodo–Coullet–Tresser maps $F (x, y, z) = (a x - b (y - z), b x + a (y - z), c x - d x k + e z)$ where $a, b, c, d, e$ are real parameters with $b d \neq 0$ and $k > 1$ is an integer. We find regions of parameters near anti-integrable limits and near singularities for which there exist hyperbolic invariant sets such that the restriction of $F$ to these sets is conjugate to the full shift on two or three symbols.