A uniform continuity condition which is equivalent to Coleman's conjecture

Abstract

At the Fifth International Conference on Nonlinear Oscillations (Kiev, 1970), Coleman [l] raised the question of whether or not a flow with a topologically hyperbolic stationary point is necessarily locally conjugate with some flow near a differentiably hyperbolic (generic) stationary point. We shall prove an affirmative result in the case where either the stable manifold or the unstable manifold has dimension one and we shall prove the equivalence of a uniform continuity condition in the general case. These affirmative results are contrasted with the recent counterexample by Neumann [2], which has both the stable manifold and the unstable manifold with dimension 2. Using Hartman’s theorem [3], it is easy to show that every differentiably hyperbolic stationary point on [w N is locally conjugate to the flow F~;,,~-,~ of the Standard Example, where