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Shadowing and the basins of terminal chain components
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- Journal:
- Canadian Mathematical Bulletin / Volume 68 / Issue 1 / March 2025
- Published online by Cambridge University Press:
- 10 December 2024, pp. 187-197
- Print publication:
- March 2025
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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_\delta $-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.
CHAIN COMPONENTS WITH THE STABLE SHADOWING PROPERTY FOR C1 VECTOR FIELDS
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- Journal:
- Journal of the Australian Mathematical Society / Volume 110 / Issue 2 / April 2021
- Published online by Cambridge University Press:
- 01 February 2021, pp. 243-259
- Print publication:
- April 2021
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Let M be a closed n-dimensional smooth Riemannian manifold, and let X be a
$C^1$-vector field of 
$M.$ Let 
$\gamma $ be a hyperbolic closed orbit of 
$X.$ In this paper, we show that X has the 
$C^1$-stably shadowing property on the chain component 
$C_X(\gamma )$ if and only if 
$C_X(\gamma )$ is the hyperbolic homoclinic class.
