Communications of the
Korean Mathematical Society CKMS

In this study, we show that the strong exponential limit shadowing property (SELmSP, for short), which has been recently introduced, exists on a neighborhood of a hyperbolic set of a diffeomorphism. We also prove that $\Omega$-stable diffeomorphisms and $\mathcal{\mathcal{L}}$-hyperbolic homeomorphisms have this type of shadowing property. By giving examples, it is shown that this type of shadowing is different from the other shadowings, and the chain transitivity and chain mixing are not necessary for it. Furthermore, we extend this type of shadowing property to positively expansive maps with the shadowing property.

Keywords: Exponential limit shadowing, limit shadowing, hyperbolic set, $\Omega$-stability, positively expansive map

MSC numbers: Primary 37C50; Secondary 37D05

Supported by: The author was supported by grant agreement No.SCU. MM1400.418 from SCU (Shahid Chamran University of Ahvaz) in Iran.