ON PIECEWISE MONOTONE FUNCTIONS WITH HEIGHT BEING INFINITY

Hong Zhu; Lin Li; Yingying Zeng; Zhiheng Yu

doi:10.11948/20210004

2021 Volume 11 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(2): 1062-1073

Next Article Previous Article

Hong Zhu, Lin Li, Yingying Zeng, Zhiheng Yu. ON PIECEWISE MONOTONE FUNCTIONS WITH HEIGHT BEING INFINITY[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 1062-1073. doi: 10.11948/20210004

Citation:

Hong Zhu, Lin Li, Yingying Zeng, Zhiheng Yu. ON PIECEWISE MONOTONE FUNCTIONS WITH HEIGHT BEING INFINITY[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 1062-1073. doi: 10.11948/20210004

ON PIECEWISE MONOTONE FUNCTIONS WITH HEIGHT BEING INFINITY

1.
College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China
2.
School of Mathematical Sciences and V.C. & V.R. Key Lab of Sichuan Province, Sichuan Normal University, Chengdu, Sichuan 610068, China
3.
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

Corresponding author: Email address: matlinl@zjxu.edu.cn (L. Li)

Abstract

Abstract

It is known that every piecewise monotone function with height finity has a characteristic interval after finite times iteration, and then the study of dynamics for such functions is able to be restricted to their characteristic intervals, which becomes monotone case. To the opposite, the description for piecewise monotone functions with height being infinity is much more complicated since the theory of characteristic interval does not work anymore. In this paper, we consider the problem of topological conjugacy for piecewise monotone functions with height being infinity. Some necessary and sufficient conditions are given for the existence of conjugacies between these functions. Moreover, the height of infinity under composition is also discussed. The fact shows a kind of symmetry for the height.
- Piecewise monotone function /
- height /
- topological conjugacy /
- fort
MSC: 39B12, 37E05, 54H20, 68W30

FullText(HTML)

References

[1]	J. Alves and J. Ramos, One-dimensional semiconjugacy revisited, Internat. J. Bifur. Chaos App. Sci. Engry., 2003, 13, 1657-1663. doi: 10.1142/S0218127403007503 CrossRef Google Scholar
[2]	S. Baldwin, A complete classification of the piecewise monotone functions of the interval, Trans. Amer. Math. Soc., 1990, 319, 155-178. doi: 10.1090/S0002-9947-1990-0961618-9 CrossRef Google Scholar
[3]	L. Block, J. Keesling and D. Ledis, Semi-conjugacies and inverse limit spaces, J. Difference Equ. Appl., 2012, 18, 627-645. doi: 10.1080/10236198.2011.611803 CrossRef Google Scholar
[4]	I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994. Google Scholar
[5]	P. Gianni, B. Trager and G. Zacharias, Gröbner bases and primary decomposition of polynomials, J. Symbolic Comput., 1988, 6, 146-167. Google Scholar
[6]	J. Guckenheimer, Sensitive dependence on initial conditions for a one dimensional maps, Commun. Math. Phys., 1979, 70, 133-160. doi: 10.1007/BF01982351 CrossRef Google Scholar
[7]	Z. Leśniak and Y. G. Shi, Topological conjugacy of piecewise monotonic functions of nonmonotonicity height $\geq 1$, J. Math. Anal. Appl., 2015, 423, 1792-1803. doi: 10.1016/j.jmaa.2014.10.063 CrossRef Google Scholar
[8]	L. Li, A topological classification for piecewise monotone iterative roots, Aequationes Math., 2017, 91, 137-152. doi: 10.1007/s00010-016-0457-4 CrossRef Google Scholar
[9]	L. Liu, W. Jarczyk, L. Li and W. Zhang, Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than $2$, Nonlinear Anal., 2012, 75, 286-303. doi: 10.1016/j.na.2011.08.033 CrossRef Google Scholar
[10]	X. Liu, Z. Yu and W. Zhang, Conjugation of rational functions to power functions and applications to iteration, Results Math., 2018, 73, 31. doi: 10.1007/s00025-018-0801-1 CrossRef Google Scholar
[11]	W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin, 1992. Google Scholar
[12]	J. Milnor and W. Thurston, On iterated maps of the interval, in: Dynamical Systems: Proceedings 1986-1987, Lectures Notes in Mathematics, Vol. 1342, Springer-Verlag, Berlin, 465-563, 1988. Google Scholar
[13]	M. Misiurewicz and S. Roth, No semiconjugacy to a map of constant slope, Ergod. Theory Dyn. Sys., 2016, 36, 875-889. doi: 10.1017/etds.2014.81 CrossRef Google Scholar
[14]	D. Ou and K. Palmer, A constructive proof of the existence of a semi-conjugacy for a one dimensional map, Discrete. Contin. Dyn. Sys. Ser. B, 2012, 17, 977-992. Google Scholar
[15]	W. Parry, Symbolic dynamics and transformations of unit interval, Trans. Amer. Math. Soc., 1966, 122, 368-378. doi: 10.1090/S0002-9947-1966-0197683-5 CrossRef Google Scholar
[16]	V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: a Computational Algebra Approach, Birkhäuser, Boston, 2009. Google Scholar
[17]	Z. Yu and L. Liu, Complexity in iteration of polynomials, Aequat. Math., 2019, 93, 985-1007. doi: 10.1007/s00010-019-00651-y CrossRef Google Scholar
[18]	Z. Yu, L. Yang and W. Zhang, Discussion on polynomials having polynomial iterative roots, J. Symbolic Comput., 2012, 47(10), 1154-1162. doi: 10.1016/j.jsc.2011.12.038 CrossRef Google Scholar
[19]	Y. Zeng and L. Li, On non-monotonicity height of piecewise monotone functions, submitted. Google Scholar
[20]	J. Zhang and L. Yang, Discussion on iterative roots of piecewise monotone functions, in Chinese, Acta Math. Sinica, 1983, 26, 398-412. Google Scholar
[21]	W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 1997, 65, 119-128. doi: 10.4064/ap-65-2-119-128 CrossRef Google Scholar