A note on the shadowing lemma of Liao: A generalized and improved version

Abstract

The shadowing lemma of Shantao Liao asserts that a “recurrent” quasi-hyperbolic string of a $C^1$-class diffeomorphism of a closed manifold might be closed up by a periodic orbit. In this note, we further show that the string can be closed exponentially up by the periodic orbit. Moreover, this statement also holds for a $C^1$-class map which is only locally diffeomorphic.