Weakly shadowable vector fields on non-oriented surfaces

Abstract

We say that a vector field has the weakly shadowing property if for any there exists such that for every d-pseudo orbit there exists an exact orbit whose -neighbourhood containing the pseudo orbit. It is proved in Li Ming and Zhongjie Liu [Weak shadowing property for flows on oriented surfaces, Proc. Amer. Math. Soc. 145(6) (2017), pp. 2591–2605.] that vector fields in the C 1-interior of the set of vector fields on an oriented smooth closed surface having the weakly shadowing property are structurally stable. In this paper, we show that the above conclusion does not hold for non-oriented surfaces. Precisely, we construct a non- -stable vector field on the Klein bottle which has the weakly shadowing property robustly.