Non-monotonicity height of PM functions on interval
Main Article Content
Abstract
Using the piecewise monotone property, we give a full description of non-monotonicity height of PM functions with a single fort on compact interval. This method is also available for general PM functions with nitely many forts, as well as those functions dened on the whole real line. Finally, we obtain a sucient and necessary condition for the nite non-monotonicity height by characteristic interval.
Article Details
How to Cite
Zhang, P.
(2017).
Non-monotonicity height of PM functions on interval.
Acta Mathematica Universitatis Comenianae, 86(2), 287-297.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/504/472
Issue
Section
Articles
References
[1] J. Banks, V. Dragan and A. Jones, Chaos - A mathematical Introduction, Cambridge University Press, 2003.
[2] M. Kuczma, On the functional eqation 'n(x) = g(x), Ann. Polon. Math., 11(1961), 161-175.
[3] Z. Lesniak and Y. Shi, Topological conjugacy of piecewise monotonic functions of non-monotonicity height 1, J. Math. Anal. Appl., 423(2015), 1792-1803.
[4] L. Li, Number of vertices for polygonal functions under iteration, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math., 14(2)(2007), 99-109.
[5] L. Li, Topological conjugacy between PM function and its iterative roots, Bull. Malays. Math. Sci. Soc., DOI 10.1007/s40840-016-0360-0.
[6] L. Li and J. Chen, Iterative roots of piecewise monotonic functions with finite non-monotonicity height, J. Math. Anal. Appl., 411(2014), 395-404.
[7] L. Li and W. Zhang, Conjugacy between piecewise monotonic functions and their iterative roots, SCIENCE CHINA Mathematics, 59(2)(2016), 367-378.
[8] L. Liu, W. Jarczyk, L. Li and W. Zhang, Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2, Nonlinear Analysis, 75(2012), 286-303.
[9] L. Liu and W. Zhang, Non-monotonic iterative roots extended from characteristic intervals, J. Math. Anal. Appl., 378(2011), 359-373.
[10] Y. Shi, L. Li and Z. Lesniak, On conjugacy of r-modal interval maps with nonmonotonicity height equal to 1, J. Difference Equ. Appl., 19(2013), 573-584.
[11] J. Zhang and L. Yang, Discussion on iterative roots of piecewise monotone functions, Acta Math. Sinica., 26(1983), 398-412. (in Chinese)
[12] J. Zhang, L. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 26(1995), 385-405.
[13] W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65(1997), 119-128.
[2] M. Kuczma, On the functional eqation 'n(x) = g(x), Ann. Polon. Math., 11(1961), 161-175.
[3] Z. Lesniak and Y. Shi, Topological conjugacy of piecewise monotonic functions of non-monotonicity height 1, J. Math. Anal. Appl., 423(2015), 1792-1803.
[4] L. Li, Number of vertices for polygonal functions under iteration, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math., 14(2)(2007), 99-109.
[5] L. Li, Topological conjugacy between PM function and its iterative roots, Bull. Malays. Math. Sci. Soc., DOI 10.1007/s40840-016-0360-0.
[6] L. Li and J. Chen, Iterative roots of piecewise monotonic functions with finite non-monotonicity height, J. Math. Anal. Appl., 411(2014), 395-404.
[7] L. Li and W. Zhang, Conjugacy between piecewise monotonic functions and their iterative roots, SCIENCE CHINA Mathematics, 59(2)(2016), 367-378.
[8] L. Liu, W. Jarczyk, L. Li and W. Zhang, Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2, Nonlinear Analysis, 75(2012), 286-303.
[9] L. Liu and W. Zhang, Non-monotonic iterative roots extended from characteristic intervals, J. Math. Anal. Appl., 378(2011), 359-373.
[10] Y. Shi, L. Li and Z. Lesniak, On conjugacy of r-modal interval maps with nonmonotonicity height equal to 1, J. Difference Equ. Appl., 19(2013), 573-584.
[11] J. Zhang and L. Yang, Discussion on iterative roots of piecewise monotone functions, Acta Math. Sinica., 26(1983), 398-412. (in Chinese)
[12] J. Zhang, L. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 26(1995), 385-405.
[13] W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65(1997), 119-128.