Recurrence and almost periodicity on dendrites

Abdelli, Hafedh; Marzougui, Habib

doi:10.1016/j.chaos.2016.03.006

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Consider B � [A B] ⊂ X\γ(x) An Arc Let I ∈ a And Let1

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2022

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“…Since E(X) � ∪ i∈A (E(C i )\ z i 􏼈 􏼉) and A is countable, there exists i ∈ A such that E(C i ) is uncountable. By [6], eorem 1.2, C i does not have the APR property. Since r i : X ⟶ C i is a retraction, by Lemma 3, X does not have the APR property.…”

Section: Consider B � [A B] ⊂ X\γ(x) An Arc Let I ∈ a And Letmentioning

confidence: 98%

Recurrence and Chaos of Local Dendrite Maps

Hattab

2022

Journal of Mathematics

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Let X be a local dendrite, and let f be a continuous self-mapping of X . Let E X represent the subset of endpoints of X . Let A P f denote the subset of almost periodic points of f , R f be the subset of recurrent points of f , and P f be the subset of periodic points of f . In this work, it is shown that R f ¯ = A P f ¯ if and only if E X is countable. Also, we show that if E X is countable, then R f = X (respectively, R f ¯ = X ) if and only if either X = S 1 , and f is a homeomorphism topologically conjugate to an irrational rotation, or P f = X (respectively, P f ¯ = X ). In this setting, we derive that if E X is countable, then, on local dendrites ≠ S 1 , transitivity = chaos.

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Section: Consider B � [A B] ⊂ X\γ(x) An Arc Let I ∈ a And Letmentioning

confidence: 98%