A note on shadowing properties

2016/09/29 by Chu, Hahng-Yun, Goo, Dae Hwan, Ku, Se-Hyun
#37C10 #37C50 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.09439

Abstract

Let \mathfrakX¹(M) be the space of C¹-vector fields on M endowed with the C¹-topology and let Λ be an isolated set for a X∈\mathfrakX¹(M). In this paper, we directly prove that every X∈\mathfrakX¹(M) having the (asymptotic) average shadowing property in Λ has no proper attractor in Λ. Our proof is a direct version of the results by Gu and Ribeiro. We also show that every X∈\mathfrakX¹(M) having the (two-sided) limit shadowing property with a gap in Λ is topologically transitive and has the shadowing property in Λ.

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