Local Shadowing Beyond Global Shadowing: Entropy and Dense Manifold Realizations

Abstract

Shadowable points were developed to cover cases in which a local shadowing mechanism survives without a global shadowing property. We show that on every compact manifold of dimension at least two, there is a $C^0$-dense set $\mathcal{R}$ of homeomorphisms so that each $f\in \mathcal{R}$ has a transitive chain component $D$ consisting of shadowable points, although $f$, $f|_D$, and every chain recurrent class meeting a neighborhood of $D$ fail shadowing. Thus ambient pointwise tracing is neither inherited from global shadowing nor explained by a shadowing subsystem. Furthermore, on general compact spaces, arbitrary Cantor dynamics can occur as the entire set of shadowable points. We also study shadowable points through the local dynamics of chain classes and derive, under additional hypotheses, semi-horseshoes, entropy-bearing ergodic approximations of measures, and entropy flexibility.