Pruning Fronts and the Formation of Horseshoes

Let f : π → π be a homeomorphism of the plane π. We define open sets P , called pruning fronts after the work of Cvitanović [C], for which it is possible to construct an isotopy H : π × [0, 1] → π with open support contained in ⋃ n∈Z f(P ) such that H(·, 0) = f(·) and H(·, 1) = fP (·), where fP is a homeomorphism under which every point of P is wandering. Applying this construction with f being Smale’s horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behaviour. This family is a 2-dimensional analog of a 1-dimensional universal family. Stony Brook IMS Preprint #1997/4 March 1997 0 Introduction One of the main concerns in the study of dynamical systems is to understand how a family of maps passes from simple to complicated dynamic behaviour as we vary parameters. When the dynamical systems under consideration are 1-dimensional, the kneading theory of Milnor and Thurston provides a full topological understanding of the transition from simple to chaotic behaviour. In dimension 2, no such theory exists. In fact , it is not clear what restrictions should be imposed on the families under consideration in order that understanding them is not too hopeless a task. Families like the Hénon and the Lozi ones are interesting examples but they lack a defining topological characteristic analogous, for example, to saying that a 1-dimensional map is unimodal (i.e., is piecewise monotone with exactly one turning point.) In this work, we present a method of isotoping away dynamics from a homeomorphism of the plane in a controlled fashion. More precisely, if f : π → π is a homeomorphism of the plane π, we define open sets P for which there exists an isotopy H : π × [0, 1] → π with (open) support 1