Generalized HÉnon Maps and Smale Horseshoes of New Types

We study hyperbolic dynamics and bifurcations for generalized Henon maps in the form (with b, α small and γ > 4). Hyperbolic horseshoes with alternating orientation, called half-orientable horseshoes, are proved to represent the nonwandering set of the maps in certain parameter regions. We show that there are infinitely many classes of such horseshoes with respect to the local topological conjugacy. We also study transitions from the usual orientable and nonorientable horseshoes to half-orientable ones (and vice versa) as parameters vary.