Journal of the
Korean Mathematical Society JKMS

In this paper, we aim to study the relations between dynamical properties of a homeomorphism (or diffeomorphism) $f$ on a Riemannian manifold $(M,g)$ and its induced lifting map $\tilde{f}$ on the universal covering $(\widetilde{M},\tilde{g})$ of $(M,g)$. We prove that the $L$-bounded (asymptotic) average shadowing properties for a homeomorphism $f$ on $(M,g)$ and the lifting $\tilde{f}$ of $f$ on $(\widetilde{M},\tilde{g})$ are equivalent but the two-sided limit shadowing properties with a gap of each space are not equivalent. We also prove that with a gap, the notion of the unique two-sided limit shadowing property is completely obtained by the notion of product Anosov diffeomorphism. It is a generalized version of \cite[Theorem A]{C-2018}. Moreover, for the case that $M$ admits a universal covering $\widetilde{M}$ with non-positive sectional curvature, we get the set of all points that two-sided limit shadow with a gap as finding fixed points of a map defined on the space of sequences in tangent spaces. Finally we find the sufficient conditions to get points which two-sided limit shadow with a gap.

Keywords: $L$-bounded (asymptotic) average shadowing property, two-sided limit shadowing property with a gap, product Anosov diffeomorphism, fixed point

MSC numbers: Primary 37B65, 37B55, 57R10