We consider dynamical systems defined by continuous maps of an interval I of the real axis into itself. We prove that if an interval J in I contains the preimage of a periodic point of period p of a map f ∈ C 0(I, I), then the sequence of intervals f 2pn(J), n= 0, 1, 2,…, is convergent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Sharkovsky, Yu. L. Maistrenko, and E.Yu. Romanenko, Difference Equations and Their Applications, Kluwer, Dordrecht (1993).
A. N. Sharkovsky and E.Yu. Romanenko, “Difference equations and dynamical systems generated by some classes of boundary-value problems,” in: Proceedings of the Steklov Institute of Mathematics, Vol. 244 (2004), pp. 264–279.
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. S. Ramos, D. Gronau, C. Mira, L. Reich, and A. N. Sharkovsky (editors), Iteration Theory (ECIT’02), Vol. 346 (2004), pp. 107–111.
E. Yu. Romanenko, “Dynamics of neighborhoods of points under a continuous mapping of an interval,” Ukr. Mat. Zh., 57, No. 11, 534–547 (2005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 130–133, January–March, 2009.
Rights and permissions
About this article
Cite this article
Fedorenko, V.V. Asymptotics of the trajectory of an interval that contains the preimage of a periodic point. Nonlinear Oscill 12, 133–136 (2009). https://doi.org/10.1007/s11072-009-0066-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11072-009-0066-4