Non-periodic bifurcations of one-dimensional maps

Abstract

We prove that a 'positive probability' subset of the boundary of '{uniformly expanding circle transformations}' consists of Kupka-Smale maps. More precisely, we construct an open class of two-parameter families of circle maps (f(alpha,theta))(alpha,theta) such that, for a positive Lebesgue measure subset of values of alpha, the family (f(alpha,theta))(theta) crosses the boundary of the uniformly expanding domain at a map for which all periodic points are hyperbolic (expanding) and no critical point is pre-periodic. Furthermore, these maps admit an absolutely continuous invariant measure. We also provide information about the geometry of the boundary of the set of hyperbolic maps.