Necessary and Sufficient Conditions for Hyperbolicity

Abstract

Sacker and Sell showed that if the linear skew product flow generated by a diffeomorphism has the so-called no nontrivial bounded solution property and if the diffeomorphism is chain recurrent on the underlying invariant set in the manifold, then this underlying set must be hyperbolic. The purpose of this note is to point out (what in fact may be well-known but the author has never seen it in print) that the assumption of chain recurrence is not necessary provided the assumption of no nontrivial bounded solution property is strengthened. This additional assumption is more or less the same as the analytic strong transversality property proved by Mañé to be equivalent to structural stability. Actually the result proved here is largely implicit in the results of Sacker and Sell but they did not state such a result explicitly. The proof here uses different techniques from those of Sacker and Sell, being based on ideas of Coppel.