On The Omega-Limit Map on 1-Dimensional Continua

• Askri, Ghassen ^[1]
  1. [1] Abdulaziz University & University of Carthage
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 3, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00511-8
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• Resumen

  • Let X be a compact connected metric space and f : X → X be a continuous map. In this paper, we prove that if f has a periodic point and ω f is continuous then the almost periodic set is a finite union of cyclically permuted subcontinua of X. In particular, AP( f ) is connected whenever f has a fixed point. Also we show that for dendrites with closed endpoint set, if ω f (a) is infinite, then ω f is continuous at a if, and only if, f is equicontinuous at a. We show that the later result fails whenever ω f (a) is finite or the endpoints set is not closed. We give an example of a local dendrite map f : X → X for which ω f is continuous, f|X∞ is equicontinuous but f is not equicontinuous on the whole space X. Finally, we answer to an open question raised by Acosta and Fernández, (Equicontinuous mappings on finite trees, Fund. Math) by providing a class of dendrites on which the equicontinuity of f|X∞ imply the equicontinuity of f .

• Referencias bibliográficas
  • 1. Abdelli, H., Askri, G.: Chain recurrent points of monotone graph maps and CR-Pm property on local dendrites. J. Diff. Equ. Appl. 25(1),...
  • 2. Acosta, G., Fernández, D.: Equicontinuous mappings on finite trees. Fund. Math. 254(2), 215–240 (2021)
  • 3. Arévalo, D., Charatonik, W.J., Pellicer-Covarrubias, P., Simn, L.: Dendrites with a closed set of endpoints. Top. App. 115, 1–17 (2001)
  • 4. Askri, G.: Equicontinuity and Li-Yorke pairs of dendrite maps. Dyn. Syst. Int. J. 35(4), 597–608 (2020)
  • 5. Askri, G., Naghmouchi, I.: Pointwise recurrence on local dendrites. Qualitative Theory Dyn. Syst. 19(1), 629–637 (2020)
  • 6. Auslander, J., Katznelson, Y.: Continuous maps of the circle without periodic points. Isr. J. Math. 32(4), 375–381 (1979)
  • 7. Block, L.S., Coppel, W.A.: Dynamics in One Dimension, Lecture Notes in Math., vol. 1513. Springer (1992)
  • 8. Blokh, A.M.: Pointwise-recurrent maps on uniquely arcwise connected locally arcwise connected spaces. Proc. AMS 143(9), 3985–4000 (2015)
  • 9. Bruckner, A.M., Hu, T.: Equicontinuity of iterates of an interval map. Tamkang. J. Math. 21, 287–294 (1990)
  • 10. Camargo, J., Rincón, M., Uzcátegui, C.: Equicontinuity of maps on dendrites. Chaos Solitons Fractals 126, 1–6 (2019)
  • 11. Illanes, A., Nadler, S.B., Jr.: Hyperspaces, Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Math, vol....
  • 12. Mai, J.: The structure of equicontinuous maps. Trans. AMS 355(10), 4125–4136 (2003)
  • 13. Mai, J., Shi, E.: R = P for maps of dendrites X with Card(End(X))
  • 14. Sun, T.X., Su, G.W., Xi, H.J.: Equicontinuity of maps on a dendrite with finite branch points. Acta Mathematica Sinica 33, 1125–1130 (2017)
  • 15. Valaristos, A.: Equicontinuity of iterates of circle maps. J. Math. Math. Sci. 2(3), 453–458 (1998)