Genetics of iterative roots for PM functions

Liu Liu; Weinian Zhang

doi:10.3934/dcds.2020369

Previous Article
Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations
DCDS Home
This Issue
Next Article
On the vanishing discount problem from the negative direction

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

Genetics of iterative roots for PM functions

Liu Liu ^1, and Weinian Zhang ^2,,

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.

Keywords: Iterative root, PM function, characteristic interval, height, topological conjugation.

Mathematics Subject Classification: Primary: 37E05, 39B12; Secondary: 26A18.

Citation: Liu Liu, Weinian Zhang. Genetics of iterative roots for PM functions. Discrete & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	L. Kindermann, Computing iterative roots with neural networks, Proc. Fifth Conf. Neural Info. Processing, 2 (1998), 713-715. Google Scholar
[8]	M. Kuczma, Functional Equations in a Single Variable, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. Google Scholar
[9]	M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9781139086639. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	G. Targoński, Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981. Google Scholar
[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. Google Scholar
[16]	J. Zhang and L. Yang, Iterative roots of a piecewise monotone continuous self-mapping, Acta. Math. Sinica, 26 (1983), 398-412. Google Scholar
[17]	J. Zhang, L. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 24 (1995), 385-405. Google Scholar
[18]	W. Zhang, A generic property of globally smooth iterative roots, Sci. China Ser. A, 38 (1995), 267-272. Google Scholar
[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. Google Scholar

show all references

References:

[1]	Ch. Babbage, Essay towards the calculus of functions, Philosoph. Transact., 105 (1815), 389-424. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	L. Kindermann, Computing iterative roots with neural networks, Proc. Fifth Conf. Neural Info. Processing, 2 (1998), 713-715. Google Scholar
[8]	M. Kuczma, Functional Equations in a Single Variable, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. Google Scholar
[9]	M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9781139086639. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	G. Targoński, Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981. Google Scholar
[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. Google Scholar
[16]	J. Zhang and L. Yang, Iterative roots of a piecewise monotone continuous self-mapping, Acta. Math. Sinica, 26 (1983), 398-412. Google Scholar
[17]	J. Zhang, L. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 24 (1995), 385-405. Google Scholar
[18]	W. Zhang, A generic property of globally smooth iterative roots, Sci. China Ser. A, 38 (1995), 267-272. Google Scholar
[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. Google Scholar

[1]	Fangzhou Cai, Song Shao. Topological characteristic factors along cubes of minimal systems. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5301-5317. doi: 10.3934/dcds.2019216
[2]	Ekaterina Gromova, Ekaterina Marova, Dmitry Gromov. A substitute for the classical Neumann–Morgenstern characteristic function in cooperative differential games. Journal of Dynamics & Games, 2020, 7 (2) : 105-122. doi: 10.3934/jdg.2020007
[3]	Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[4]	José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781
[5]	Silvére Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik. Computability of topological entropy: From general systems to transformations on Cantor sets and the interval. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4259-4286. doi: 10.3934/dcds.2020180
[6]	Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[7]	Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089
[8]	Tadeusz Antczak. The $ F $-objective function method for differentiable interval-valued vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2761-2782. doi: 10.3934/jimo.2020093
[9]	Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191
[10]	Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008
[11]	Andrew L. Nevai, Richard R. Vance. The role of leaf height in plant competition for sunlight: analysis of a canopy partitioning model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 101-124. doi: 10.3934/mbe.2008.5.101
[12]	Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911
[13]	Gabriele Beltramo, Primoz Skraba, Rayna Andreeva, Rik Sarkar, Ylenia Giarratano, Miguel O. Bernabeu. Euler characteristic surfaces. Foundations of Data Science, 2021 doi: 10.3934/fods.2021027
[14]	Santanu Sarkar, Subhamoy Maitra. Some applications of lattice based root finding techniques. Advances in Mathematics of Communications, 2010, 4 (4) : 519-531. doi: 10.3934/amc.2010.4.519
[15]	Vincenzo Ambrosio, Giovanni Molica Bisci, Dušan Repovš. Nonlinear equations involving the square root of the Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 151-170. doi: 10.3934/dcdss.2019011
[16]	Laurenţiu Maxim, Jörg Schürmann. Characteristic classes of singular toric varieties. Electronic Research Announcements, 2013, 20: 109-120. doi: 10.3934/era.2013.20.109
[17]	Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[18]	Yan Liu, Minjia Shi, Hai Q. Dinh, Songsak Sriboonchitta. Repeated-root constacyclic codes of length $ 3\ell^mp^s $. Advances in Mathematics of Communications, 2020, 14 (2) : 359-378. doi: 10.3934/amc.2020025
[19]	Partha Sharathi Dutta, Soumitro Banerjee. Period increment cascades in a discontinuous map with square-root singularity. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 961-976. doi: 10.3934/dcdsb.2010.14.961
[20]	Tingting Wu, Shixin Zhu, Li Liu, Lanqiang Li. Repeated-root constacyclic codes of length 6l^mpⁿ. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021044

American Institute of Mathematical Sciences

Genetics of iterative roots for PM functions

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Genetics of iterative roots for PM functions

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors