The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C¹ interior of the set of all diffeomorphisms on a closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C¹ interior is finite if and only if f is Morse-Smale.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm168-1-2,
author = {Kazuhiro Sakai},
title = {Diffeomorphisms with weak shadowing},
journal = {Fundamenta Mathematicae},
volume = {167},
year = {2001},
pages = {57-75},
zbl = {0968.37009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm168-1-2}
}
Kazuhiro Sakai. Diffeomorphisms with weak shadowing. Fundamenta Mathematicae, Tome 167 (2001) pp. 57-75. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm168-1-2/