Tohoku Mathematical Journal
2010
December
SECOND SERIES VOL. 62, NO. 4

Tohoku Math. J.
62 (2010), 509-526

Title A NOTE ON THE SHADOWING LEMMA OF LIAO: A GENERALIZED AND IMPROVED VERSION

Author Xiongping Dai

(Received August 10, 2009, revised February 9, 2010)
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.

2000 Mathematics Subject Classification. Primary 37C50; Secondary 37D30.
Key words and phrases. Shadowing lemma, exponential closing property, quasi-hyperbolicity, local diffeomorphism.

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