-
Previous Article
Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables
- DCDS Home
- This Issue
-
Next Article
Integral equations on compact CR manifolds
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 |
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.
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. |
[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. Google Scholar |
[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. |
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. |
[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. Google Scholar |
[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. |
[1] |
Sergio Zamora. Tori can't collapse to an interval. Electronic Research Archive, , () : -. doi: 10.3934/era.2021005 |
[2] |
Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020278 |
[3] |
Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350 |
[4] |
Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 |
[5] |
Tingting Wu, Li Liu, Lanqiang Li, Shixin Zhu. Repeated-root constacyclic codes of length $ 6lp^s $. Advances in Mathematics of Communications, 2021, 15 (1) : 167-189. doi: 10.3934/amc.2020051 |
[6] |
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 |
[7] |
Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020443 |
[8] |
Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117 |
[9] |
Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021 doi: 10.3934/nhm.2021004 |
[10] |
Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017 |
[11] |
Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 |
[12] |
Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021011 |
[13] |
Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020119 |
[14] |
Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]