Shadowing and the basins of terminal chain components
Noriaki Kawaguchi
TL;DR
The paper investigates how the shadowing property constrains the global dynamics of a continuous map on a compact metric space, focusing on the basins of terminal chain components and the potential for almost chain continuity. It provides a constructive proof that the sets $V(f)$ and $W(f)$, capturing points tied to terminal components and to shadowing-compatible behavior, are dense $G_\delta$-subsets of $X$, by building explicit dense $G_\delta$ sets $V$ and $W$ and showing their equality to $V(f)$ and $W(f)$ respectively. Under local connectedness of $X$ and total disconnectedness of the chain recurrent set $CR(f)$, the work proves that the map is almost chain continuous, linking shadowing to topological regularity in generic settings. The results extend and offer an alternative route to known facts about generic dynamics on manifolds and related spaces (e.g., dendrites), highlighting the role of terminal chain components in organizing global behavior. Overall, the paper strengthens the relationship between shadowing, terminal components, and chain-continuous dynamics, with implications for the structure of attractor-like sets and for understanding generic properties of dynamical systems.
Abstract
We provide an alternative view of some results in [1, 3, 11]. In particular, we prove that (1) if a continuous self-map of a compact metric space has the shadowing, then the union of the basins of terminal chain components is a dense $G_δ$-subset of the space; and (2) if a continuous self-map of a locally connected compact metric space has the shadowing, and if the chain recurrent set is totally disconnected, then the map is almost chain continuous.