We consider dynamical systems generated by continuous mappings of an interval I into itself. We prove that the trajectory of an interval J ⊂ I is asymptotically periodic if and only if J contains an asymptotically periodic point.
Similar content being viewed by others
References
A. N. Sharkovsky, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1986); English translation: Kluwer, Dordrecht (1993).
A. N. Sharkovsky and E. Yu. Romanenko, “Difference equations and dynamical systems generated by some classes of boundaryvalue problems,” Proc. Steklov Inst. Math., 244, 264–279 (2004).
F. Hausdorff, Theory of Sets [Russian translation], ONTI NKTI SSSR, Moscow (1937).
K. Kuratowski, Topology [Russian translation], Vol. 1, Mir, Moscow (1966).
V. V. Fedorenko, “Topological limit of trajectories of intervals of simplest one-dimensional dynamical systems,” Ukr. Mat. Zh., 54, No. 3, 425–430 (2002).
V. V. Fedorenko, “Topological limit of trajectories of intervals of one-dimensional dynamical systems,” in: J. Sousa Ramos, D. Gronau, C. Mira, et al. (editors), Iteration Theory (ECIT’02) (Grazer Math. Ber., Bericht Nr.), Vol. 346 (2004), pp. 107–111.
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dynamics of One-Dimensional Maps [in Russian], Naukova Dumka, Kiev (1989); English translation: Kluwer, Dordrecht (1997).
L. S. Block and W. A. Coppel, Dynamics in One Dimension, Springer, Berlin (1992).
V. V. Fedorenko, “Asymptotics of the trajectory of an interval that contains the preimage of a periodic point,” Nelin. Kolyvannya, 12, No. 1, 130–133 (2009).
E. Yu. Romanenko, “Dynamics of neighborhoods of points under a continuous mapping of an interval,” Ukr. Mat. Zh., 57, No. 11, 1534–1547 (2005).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 5, pp. 716–720, May, 2009.
Rights and permissions
About this article
Cite this article
Fedorenko, V.V. Asymptotic periodicity of trajectories of an interval. Ukr Math J 61, 854–858 (2009). https://doi.org/10.1007/s11253-009-0238-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0238-5