# American Institute of Mathematical Sciences

May  2021, 41(5): 2391-2409. doi: 10.3934/dcds.2020369

## Genetics of iterative roots for PM functions

 1 Department of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China 2 Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Weinian Zhang

Received  February 2020 Revised  August 2020 Published  May 2021 Early access  November 2020

Fund Project: This research was supported by NSFC # 11831012 and # 11821001

It is known that the time-one mapping of a flow defines a discrete dynamical system with the same dynamical behaviors as the flow, but conversely one wants to know whether a flow embedded by a homeomorphism preserves the dynamical behaviors of the homeomorphism. In this paper we consider iterative roots, a weak version of embedded flows, for the preservation. We refer an iterative root to be genetic if it is topologically conjugate to its parent function. We prove that none of PM functions with height being $>1$ has a genetic root and none of iterative roots of height being $>1$ is genetic even if the height of its parent function is equal to 1. This shows that most functions do not have a genetic iterative root. Further, we obtain a necessary and sufficient conditions under which a PM function $f$ has a genetic iterative root in the case that $f$ and the iterative root are both of height 1.

Citation: Liu Liu, Weinian Zhang. Genetics of iterative roots for PM functions. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2391-2409. doi: 10.3934/dcds.2020369
##### References:
 [1] Ch. Babbage, Essay towards the calculus of functions, Philosoph. Transact., 105 (1815), 389-424. [2] K. Baron and W. Jarczyk, Recent results on functional equations in a single variable, perspectives and open problems, Aequationes Math., 61 (2001), 1-48.  doi: 10.1007/s000100050159. [3] A. Blokh, E. Coven, M. Misiurewicz and Z. Nitecki, Roots of continuous piecewise monotone maps of an interval, Acta Math. Univ. Comenian.(N.S.), 60 (1991), 3-10. [4] M. K. Fort, Jr. The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960–967. doi: 10.1090/S0002-9939-1955-0080911-2. [5] N. Iannella and L. Kindermann, Finding iterative roots with a spiking neural network, Inform. Process. Lett., 95 (2005), 545-551.  doi: 10.1016/j.ipl.2005.05.022. [6] S. Karlin and J. McGregor, Embeddablility of discrete time simple branching processes into continous time branching processes, Trans. Amer. Math. Soc., 132 (1968), 115-136.  doi: 10.1090/S0002-9947-1968-0222966-1. [7] L. Kindermann, Computing iterative roots with neural networks, Proc. Fifth Conf. Neural Info. Processing, 2 (1998), 713-715. [8] M. Kuczma, Functional Equations in a Single Variable, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. [9] M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639. [10] L. Li, D. Yang and W. Zhang, A note on iterative roots of PM functions, J. Math. Anal. Appl., 341 (2008), 1482-1486.  doi: 10.1016/j.jmaa.2007.11.006. [11] L. Li and W. Zhang, Conjugacy between piecewise monotonic functions and their iterative roots, Sci. China Math., 59 (2016), 367-378.  doi: 10.1007/s11425-015-5065-6. [12] L. Liu and W. Zhang, Non-monotonic iterative roots extended from characteristic interval, J. Math. Anal. Appl., 378 (2011), 359-373.  doi: 10.1016/j.jmaa.2011.01.037. [13] L. Liu, W. Jarczyk, L. Li and W. Zhang, Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2, Nonlinear Anal., 75 (2012), 286-303.  doi: 10.1016/j.na.2011.08.033. [14] G. Targoński, Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981. [15] M. C. Zdun and W. Zhang, Koenigs embedding flow problem with global $C^1$ smoothness, J. Math. Anal. Appl., 374 (2011), 633-643.  doi: 10.1016/j.jmaa.2010.08.075. [16] J. Zhang and L. Yang, Iterative roots of a piecewise monotone continuous self-mapping, Acta. Math. Sinica, 26 (1983), 398-412. [17] J. Zhang, L. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 24 (1995), 385-405. [18] W. Zhang, A generic property of globally smooth iterative roots, Sci. China Ser. A, 38 (1995), 267-272. [19] W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65 (1997), 119-128.  doi: 10.4064/ap-65-2-119-128.

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##### References:
 [1] Ch. Babbage, Essay towards the calculus of functions, Philosoph. Transact., 105 (1815), 389-424. [2] K. Baron and W. Jarczyk, Recent results on functional equations in a single variable, perspectives and open problems, Aequationes Math., 61 (2001), 1-48.  doi: 10.1007/s000100050159. [3] A. Blokh, E. Coven, M. Misiurewicz and Z. Nitecki, Roots of continuous piecewise monotone maps of an interval, Acta Math. Univ. Comenian.(N.S.), 60 (1991), 3-10. [4] M. K. Fort, Jr. The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960–967. doi: 10.1090/S0002-9939-1955-0080911-2. [5] N. Iannella and L. Kindermann, Finding iterative roots with a spiking neural network, Inform. Process. Lett., 95 (2005), 545-551.  doi: 10.1016/j.ipl.2005.05.022. [6] S. Karlin and J. McGregor, Embeddablility of discrete time simple branching processes into continous time branching processes, Trans. Amer. Math. Soc., 132 (1968), 115-136.  doi: 10.1090/S0002-9947-1968-0222966-1. [7] L. Kindermann, Computing iterative roots with neural networks, Proc. Fifth Conf. Neural Info. Processing, 2 (1998), 713-715. [8] M. Kuczma, Functional Equations in a Single Variable, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. [9] M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639. [10] L. Li, D. Yang and W. Zhang, A note on iterative roots of PM functions, J. Math. Anal. Appl., 341 (2008), 1482-1486.  doi: 10.1016/j.jmaa.2007.11.006. [11] L. Li and W. Zhang, Conjugacy between piecewise monotonic functions and their iterative roots, Sci. China Math., 59 (2016), 367-378.  doi: 10.1007/s11425-015-5065-6. [12] L. Liu and W. Zhang, Non-monotonic iterative roots extended from characteristic interval, J. Math. Anal. Appl., 378 (2011), 359-373.  doi: 10.1016/j.jmaa.2011.01.037. [13] L. Liu, W. Jarczyk, L. Li and W. Zhang, Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2, Nonlinear Anal., 75 (2012), 286-303.  doi: 10.1016/j.na.2011.08.033. [14] G. Targoński, Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981. [15] M. C. Zdun and W. Zhang, Koenigs embedding flow problem with global $C^1$ smoothness, J. Math. Anal. Appl., 374 (2011), 633-643.  doi: 10.1016/j.jmaa.2010.08.075. [16] J. Zhang and L. Yang, Iterative roots of a piecewise monotone continuous self-mapping, Acta. Math. Sinica, 26 (1983), 398-412. [17] J. Zhang, L. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 24 (1995), 385-405. [18] W. Zhang, A generic property of globally smooth iterative roots, Sci. China Ser. A, 38 (1995), 267-272. [19] W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65 (1997), 119-128.  doi: 10.4064/ap-65-2-119-128.
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