A characterization of <i>ω</i>-leaf-limit sets for systems with foliations

The main result of this paper is that a dynamical system $ f:M\rightarrow M $ with a continuous and uniformly compact foliation $ W $ which has quasi-shadowing or leaf-shadowing has the property that the closure of $ \omega^{W}(f) $ in the Hausdorff metric coincides with $ \text{ILCT}(f) $, where $ \omega^{W}(f) $ is the collection of $ \omega $-leaf-limit sets of $ f $ and $ \text{ILCT}(f) $ is the collection of closed internally leaf-chain transitive subsets. Meanwhile, we characterize the equality $ \overline{\omega^{W}(f)} = \text{ILCT}(f) $ in terms of certain orbital leaf-shadowing. Moreover, we also prove that limit quasi-shadowing or limit leaf-shadowing implies $ \omega^{W}(f) = \text{ILCT}(f) $, which holds if and only if $ f $ satisfies certain orbital limit leaf-shadowing.