Henon-like maps with strange attractors: there exist C∞ Kupka-Smale diffeomorphisms on S2 with neither sinks nor sources

Abstract

The authors prove the existence of Cinfinity Kupka-Smale diffeomorphisms on S2 with no periodic sinks or sources. Thus they obtain a complete answer to a question raised by Smale in 1962. Bowen, Franks and Young (1976) have previously constructed C2 examples. Their construction shows that in generic Cinfinity one-parameter families there exist maps which have a non-hyperbolic strange attractor. These families include the Henon family (x,y) to (1-ax2+y,bx). (In the statements of the theorems they describe what they mean by a strange attractor). Also, as a by-product of their method, they get the first examples of Cinfinity diffeomorphisms which are both at the accumulation of a cascade of period-doubling bifurcations and at the boundary of Morse-Smale systems. They show that all these maps have a non-wandering set which is conjugate to the non-wandering set of the corresponding one-dimensional map.