非特征端点条件下PM函数的迭代根

Abstract

Abstract:

For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. This raises an open problem about iterative roots without this condition, called characteristic endpoints problem. This problem is solved almost completely when the number of forts is equal to or less than the order. In this paper, we study the case that the number of forts is greater than the order and give a sufficient condition for existence of continuous iterative roots of order 2 with height 2, answering the characteristic endpoints problem partially.

Key words: Iterative root, Piecewise monotone function, Characteristic interval, Characteristic endpoints condition

CLC Number:

O175

Xiao Tang,Lin Li. Characteristic Endpoints Question for Piecewise Monotone Functions[J].Acta mathematica scientia,Series A, 2022, 42(2): 557-569.

Trendmd

Figures/Tables 4

References 17

1	Babbage C . An Essay towards the calculus of functions. Philos Trans R Soc Lond, 1815, 105: 389- 423
2	Baron K , Jarczyk W . Recent results on functional equations in a single variable, perspectives and open problems. Aequationes Math, 2001, 61: 1- 48 doi: 10.1007/s000100050159
3	Fort M . The embedding of homeomorphisms in flows. Proc Amer Math Soc, 1955, 6: 960- 967 doi: 10.1090/S0002-9939-1955-0080911-2
4	Irwin M C . Smooth Dynamical Systems. New York: Academic Press, 1980
5	Kuczma M . On the functional equation $\varphi^n (x) = g(x)$. Ann Polon Math, 1961, 11: 161- 175 doi: 10.4064/ap-11-2-161-175
6	Kuczma M . Functional Equations in a Single Variable. Warszawa: Polish Scientific Publishers, 1968
7	Kuczma M , Choczewski B , Ger R . Iterative Functional Equations. Cambridge: Cambridge University Press, 1990
8	Li L , Chen J . Iterative roots of piecewise monotonic functions with finite nonmonotonicity height. J Math Anal Appl, 2014, 411: 395- 404 doi: 10.1016/j.jmaa.2013.09.051
9	Li L , Yang D L , Zhang W N . A note on iterative roots of PM functions. J Math Anal Appl, 2008, 341: 1482- 1486 doi: 10.1016/j.jmaa.2007.11.006
10	Li L , Zhang W N . The question on characteristic endpoints for iterative roots of PM functions. J Math Anal Appl, 2018, 458: 265- 280 doi: 10.1016/j.jmaa.2017.09.009
11	Liu L , Jarczyk W , Li L , Zhang W N . 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
12	Palis J , Melo W . Geometric Theory of Dynamical Systems: An Introduction. New York: Springer, 1982
13	Solarz P . General theorem for the existence of iterative roots of homeomorphisms with periodic points. J Math Anal Appl, 2012, 394: 391- 399 doi: 10.1016/j.jmaa.2012.03.048
14	Thompson J M T , Stewart H B . Nonlinear Dynamics and Chaos. Chichester: John Wiley & Sons, 1986
15	Zdun M , Solarz P . Recent results on iteration theory: iteration groups and semigroups in the real case. Aequationes Math, 2014, 87: 201- 245 doi: 10.1007/s00010-013-0186-x
16	张景中, 杨路. 论逐段单调连续函数的迭代根. 数学学报, 1983, 26 (4): 398- 412
17	Zhang W N . PM functions, their characteristic intervals and iterative roots. Ann Polon Math, 1997, 65: 119- 128 doi: 10.4064/ap-65-2-119-128

阶数	$ f $的性质	存在性或不存在性
$ n= 2 $	$ f $在$ [a', b'] $上$ \uparrow $, $ H(f)=1 $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)=1 $	存在
$ 2<n<N(F) $	$ f $在$ [a', b'] $上$ \uparrow $, $ H(f)<n $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)<n-1 $	不存在
$ n=N(F) $	$ f $在$ [a', b'] $上$ \uparrow $, $ H(f)\le n $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)<n-1 $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)=n $	存在
$ n=N(F)+1 $	$ f $在$ [a', b'] $上$ \uparrow $, $ H(f)\le n $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)<n-1 $	不存在

Characteristic Endpoints Question for Piecewise Monotone Functions

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 4

References 17

Related Articles 1

Metrics

Comments

Recommended 10