DENSITY OF PERIODIC ORBITS IN ω-LIMIT SETS WITH THE HAUSDORFF METRIC

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.

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The ω-limit sets of a flow and periodic orbits

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2008.10.004 ◽

2009 ◽

Vol 41 (5) ◽

pp. 2690-2696

Author(s):

Xiaoxia Wang ◽

Denis Blackmore ◽

Chengwen Wang

Keyword(s):

Periodic Orbits ◽

Limit Sets

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Invariant Sets for Monotone Local Dendrite Maps

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501509 ◽

2016 ◽

Vol 26 (09) ◽

pp. 1650150 ◽

Cited By ~ 3

Author(s):

Hafedh Abdelli ◽

Habib Marzougui

Keyword(s):

Unit Circle ◽

Periodic Point ◽

Hausdorff Metric ◽

Irrational Rotation ◽

The Other ◽

Invariant Sets ◽

Limit Sets ◽

Cut Point ◽

Monotone Map ◽

The Family

Let [Formula: see text] be a local dendrite and let [Formula: see text] be a monotone map. Denote by [Formula: see text], RR[Formula: see text], UR[Formula: see text], [Formula: see text] the set of periodic (resp., regularly recurrent, uniformly recurrent, recurrent) points and [Formula: see text] the union of all [Formula: see text]-limit sets of [Formula: see text]. We show that if [Formula: see text] is nonempty, then (i) [Formula: see text]. (ii) R[Formula: see text] if and only if every cut point is a periodic point. If [Formula: see text] is empty, then (iii) [Formula: see text]. (iv) R[Formula: see text] if and only if [Formula: see text] is a circle and [Formula: see text] is topologically conjugate to an irrational rotation of the unit circle [Formula: see text]. On the other hand, we prove that [Formula: see text] has no Li–Yorke pair. Moreover, we show that the family of all [Formula: see text]-limit sets of [Formula: see text] is closed with respect to the Hausdorff metric.

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