October  2022, 42(10): 4965-4990. doi: 10.3934/dcds.2022082

Iterative roots of type $ \mathcal {T}_2 $

1. 

Department of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

2. 

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

3. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

*Corresponding author: Weinian Zhang

Received  December 2021 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: The fourth author is supported by NSFC grant #11831012, #11821001 and #12171336

This paper aims to an open problem on iterative roots of PM functions, a class of non-monotonic functions. The open problem asks: Does a PM function of nonmonotonicity height $ \ge 2 $ have a continuous iterative root of order $ n $ being less than or equal to the number of forts? It was proved that iterative roots of order $ n $ being equal to the number of forts (if exist) can be classified into two types: mostly increasing ones (type $ \mathcal {T}_1 $) and mostly decreasing ones (type $ \mathcal {T}_2 $) and all roots of type $ \mathcal {T}_1 $ are found, but the remaining type $ \mathcal {T}_2 $ is left for more complicated construction. In this paper full description of type $ \mathcal {T}_2 $ roots is given.

Citation: Liu Liu, Justyna Jarczyk, Witold Jarczyk, Weinian Zhang. Iterative roots of type $ \mathcal {T}_2 $. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4965-4990. doi: 10.3934/dcds.2022082
References:
[1]

Jr. M. K. Fort, The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960-967.  doi: 10.1090/S0002-9939-1955-0080911-2.

[2]

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.

[3] M. C. Irwin, Smooth Dynamical Systems, Academic Press, 1980. 
[4]

L. Kindermann, Computing iterative roots with neural networks, Proc. Fifth Conf. Neural Info. Processing, 2 (1998), 713-715. 

[5]

M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publ., Warsaw, 1968.

[6] M. KuczmaB. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639.
[7]

L. LiuW. JarczykL. 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.

[8]

L. LiuL. Li and W. Zhang, Open question on lower order iterative roots for PM functions, J. Diff. Equ. Appl., 24 (2018), 825-847.  doi: 10.1080/10236198.2018.1437152.

[9]

L. Liu and W. Zhang, Non-monotonic iterative roots extended from characteristic intervals, J. Math. Anal. Appl., 378 (2011), 359-373.  doi: 10.1016/j.jmaa.2011.01.037.

[10]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An introduction, Springer, New York, 1982.

[11]

T. Sun, Iterative roots of anti-N-type functions on intervals, [Chinese], J. Math. Study, 33 (2000), 274-280. 

[12]

T. Sun and H. Xi, Iterative roots of $N$-type of functions on intervals, [Chinese], J. Math. Study, 29 (1996), 40-45. 

[13]

G. Targonski, Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981.

[14]

G. Targonski, Progress of iteration theory since 1981, Aequationes Math., 50 (1995), 50-72.  doi: 10.1007/BF01831113.

[15]

J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos, John Wiley & Sons, 1986.

[16]

G. Zhang, Conjugacy and iterative roots of a class of linear self-mapping (I), [Chinese], Chin. Ann. Math. A, 13 (1992), 33-40. 

[17]

J. Zhang and L. Yang, Iterative roots of a piecewise monotone continuous self-mapping, Acta Math. Sinica, 26 (1983), 398-412. 

[18]

J. ZhangL. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 24 (1995), 385-405. 

[19]

W. Zhang, A generic property of globally smooth iterative roots, Science in China A, 38 (1995), 267–272.

[20]

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.

show all references

References:
[1]

Jr. M. K. Fort, The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960-967.  doi: 10.1090/S0002-9939-1955-0080911-2.

[2]

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.

[3] M. C. Irwin, Smooth Dynamical Systems, Academic Press, 1980. 
[4]

L. Kindermann, Computing iterative roots with neural networks, Proc. Fifth Conf. Neural Info. Processing, 2 (1998), 713-715. 

[5]

M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publ., Warsaw, 1968.

[6] M. KuczmaB. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639.
[7]

L. LiuW. JarczykL. 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.

[8]

L. LiuL. Li and W. Zhang, Open question on lower order iterative roots for PM functions, J. Diff. Equ. Appl., 24 (2018), 825-847.  doi: 10.1080/10236198.2018.1437152.

[9]

L. Liu and W. Zhang, Non-monotonic iterative roots extended from characteristic intervals, J. Math. Anal. Appl., 378 (2011), 359-373.  doi: 10.1016/j.jmaa.2011.01.037.

[10]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An introduction, Springer, New York, 1982.

[11]

T. Sun, Iterative roots of anti-N-type functions on intervals, [Chinese], J. Math. Study, 33 (2000), 274-280. 

[12]

T. Sun and H. Xi, Iterative roots of $N$-type of functions on intervals, [Chinese], J. Math. Study, 29 (1996), 40-45. 

[13]

G. Targonski, Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981.

[14]

G. Targonski, Progress of iteration theory since 1981, Aequationes Math., 50 (1995), 50-72.  doi: 10.1007/BF01831113.

[15]

J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos, John Wiley & Sons, 1986.

[16]

G. Zhang, Conjugacy and iterative roots of a class of linear self-mapping (I), [Chinese], Chin. Ann. Math. A, 13 (1992), 33-40. 

[17]

J. Zhang and L. Yang, Iterative roots of a piecewise monotone continuous self-mapping, Acta Math. Sinica, 26 (1983), 398-412. 

[18]

J. ZhangL. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 24 (1995), 385-405. 

[19]

W. Zhang, A generic property of globally smooth iterative roots, Science in China A, 38 (1995), 267–272.

[20]

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.

Figure 1.  type $ \mathcal {T}_1 $, lower
Figure 2.  type $ \mathcal {T}_1 $, upper
Figure 3.  type $ \mathcal {T}_2 $, upper
Figure 4.  type $ \mathcal {T}_2 $, lower
Figure 5.  $ S(f) = \{c_1\} $
Figure 6.  $ S(f) = \{c_n\} $
Figure 7.  $ F $ is in (i-1)
Figure 8.  $ F $ is in (i-2)
Figure 9.  $ F $ is in (i-3)
Figure 10.  $ F $ is in (i-3)
Figure 11.  $ F $ is in (ii-1)
Figure 12.  $ F $ is in (ii-2)
Figure 13.  $ F $ is in (iii-2)
Figure 14.  $ F $ is in (iv-2)
Figure 15.  $ F_1 $ with $ N(F_1) = 3 $, $ f\in \mathcal {T}_2 $ with $ S(f) = \frac{1}{4}$
Figure 16.  $ F_2 $ with $ N(F_2) = 4 $, $ f\in \mathcal {T}_2 $ with $ S(f) = \frac{7}{8}$
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