Nonsingular vector fields in \({\mathcal G}^1(M^3)\) satisfy Axiom A and no cycle: A new proof of Liao's theorem
Let \(M\) be a compact 3-dimensional smooth manifold without boundary. Let \({\mathcal G}^1(M)\) be the set of \(C^1\) vectorfields on \(M\) which have a neighborhood \({\mathcal U}\) such that any \(Y \in {\mathcal U}\) has all its periodic orbits and singularities hyperbolic. The author proves that if \(X \in {\mathcal U}\) has no singularities, then it satisfies Axiom A and no-cycle property. The main tool in the proof was Mañé's Ergodic Closing Lemma. The author noted that the same result was proved in [\textit{S. T. Liao}, Obstruction sets. II, Acta Sci. Natur. Univ. Pekin. 2, 1-36 (1981)], by using unfamiliar techniques.
scientific article; zbMATH DE number 1448528