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Orbital shadowing, internal chain transitivity and $\unicode[STIX]{x1D714}$-limit sets

Published online by Cambridge University Press:  13 July 2016

CHRIS GOOD  and
JONATHAN MEDDAUGH
CHRIS GOOD
Affiliation:
University of Birmingham, School of Mathematics, Birmingham B15 2TT, UK email [email protected], [email protected]
JONATHAN MEDDAUGH
Affiliation:
University of Birmingham, School of Mathematics, Birmingham B15 2TT, UK email [email protected], [email protected]
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Abstract

Let $f:X\rightarrow X$ be a continuous map on a compact metric space, let $\unicode[STIX]{x1D714}_{f}$ be the collection of $\unicode[STIX]{x1D714}$-limit sets of $f$ and let $\mathit{ICT}(f)$ be the collection of closed internally chain transitive subsets. Provided that $f$ has shadowing, it is known that the closure of $\unicode[STIX]{x1D714}_{f}$ in the Hausdorff metric coincides with $\mathit{ICT}(f)$. In this paper, we prove that $\unicode[STIX]{x1D714}_{f}=\mathit{ICT}(f)$ if and only if $f$ satisfies Pilyugin’s notion of orbital limit shadowing. We also characterize those maps for which $\overline{\unicode[STIX]{x1D714}_{f}}=\mathit{ICT}(f)$ in terms of a variation of orbital shadowing.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 143 - 154
Copyright
© Cambridge University Press, 2016 

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