Topological horseshoes for Arneodo-Coullet-Tresser maps

Abstract

In this paper, we study the family of Arneodo-Coullet-Tresser maps F(x, y, z) = (ax - b(y - z), bx + a(y - z), cx - dx(k) + ez) where a, b, c, d, e are real parameters with bd not equal 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.