Abstract
In previous work, a topological conjugacy between PM functions and their iterative roots, which are increasing on the characteristic interval, was considered. In the present paper, we continue to investigate those functions that are decreasing on the characteristic interval. The differences come from the fact that neither orientation nor monotonicity can be preserved under iteration in the decreasing case. Thus, we need to partition the whole interval with a different manner and then prove the existence of topological conjugacy on each sub-interval. We also give a construction method for the conjugacies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blokh, A., Coven, E., Misiurewicz, M., Hitecki, Z.: Roots of continuous piecewise monotone maps of an interval. Acta Math. Univ. Comen. LX(1), 3–10 (1991)
Jarczyk, W.: Reversibility of interval homeomorphisms with no fixed points. Aequ. Math. 63, 66–75 (2002)
Jarczyk, W.: Reversible interval homeomorphisms. J. Math. Anal. Appl. 272, 473–479 (2002)
Kuczma, M.: Functional Equations in a Single Variable. Polish Scientific Publications, Warsaw (1968)
Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Encyclopedia of Mathematics and its Applications, vol. 32. Cambridge University Press, Cambridge (1990)
Laitochová, J.: On conjugacy equations for iterative functional equations. Int. J. Pure Appl. Math. 26, 423–433 (2006)
Leśniak, Z., Shi, Y.: Topological conjugacy of piecewise monotonic functions of nonmonotonicity height \(\ge \)1. J. Math. Anal. Appl. 423, 1792–1803 (2015)
Li, L., Yang, D., Zhang, W.: A note on iterative roots of PM functions. J. Math. Anal. Appl. 341, 1482–1486 (2008)
Li, L., Zhang, W.: On conjugacy between piecewise monotonic function and its iterative roots. Sci. China Ser. A (to appear)
Liu, L., Zhang, W.: Non-monotonic iterative roots extended from characteristic intervals. J. Math. Anal. Appl. 378, 359–373 (2011)
Liu, L., Jarczyk, W., Li, L., Zhang, W.: Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2. Nonlinear Anal. 75, 286–303 (2012)
de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (1992)
Shi, Y.: Non-monotonic solutions and continuously differentiable solutions of conjugacy equations. Appl. Math. Comput. 215, 2399–2404 (2009)
Zhang, J., Yang, L.: Discussion on iterative roots of piecewise monotone functions. Acta Math. Sin. 26, 398–412 (1983). (in Chinese)
Zhang, J., Yang, L., Zhang, W.: Some advances on functional equations. Adv. Math. Chin. 24, 385–405 (1995)
Zhang, W.: PM functions, their characteristic intervals and iterative roots. Ann. Polon. Math. 65, 119–128 (1997)
Acknowledgments
The author is grateful to the referees for their careful reading and comments. The author also thanks Professor C. T. Ng and Professor Wenmeng Zhang for their valuable suggestions. This work is supported by the National Science Foundation of China (11301226) and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ13A010017.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rosihan M. Ali.
Rights and permissions
About this article
Cite this article
Li, L. Topological Conjugacy Between PM Function and Its Iterative Roots. Bull. Malays. Math. Sci. Soc. 41, 751–764 (2018). https://doi.org/10.1007/s40840-016-0360-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-016-0360-0
Keywords
- Topological conjugacy
- Piecewise monotonic function
- Iterative root
- Non-monotonicity height
- Periodic point