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On non-monotonicity height of piecewise monotone functions

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Abstract

Non-monotonicity height is an important index to describe the complexity of dynamics for piecewise monotone functions. Although it is used extensively in the theory of iterative roots, its calculation is still difficult especially in the infinite case. In this paper, by introducing the concept of spanning interval, we first present a sufficient condition for piecewise monotone functions to have height infinity and then an algorithm for finding the spanning intervals is given. We further investigate the density of all piecewise monotone functions with infinite and finite height, respectively, and the results indicate the instability of height. At the end of this paper, the variance of height under composition, especially for functions of height 1 and infinity, are also discussed.

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Acknowledgements

The authors are very grateful to the reviewers for their carefully checking and helpful comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences and V.C. & V.R. Key Lab of Sichuan Province, Sichuan Normal University, Chengdu, 610068, Sichuan, People’s Republic of China

    Yingying Zeng

  2. Department of Mathematics, Jiaxing University, Jiaxing, 314001, Zhejiang, People’s Republic of China

    Lin Li

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  1. Yingying Zeng
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  2. Lin Li
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Correspondence to Lin Li.

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This work is supported by Zhejiang Provincial Natural Science Foundation of China under Grant #LY18A010017, the National Science Foundation of China Grants #11501394, #12026207, the Science Research Fund of Sichuan Provincial Education Department #15ZB0041 and funding of School of Mathematical Sciences and V.C. & V.R. Key Lab of Sichuan Province.

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Zeng, Y., Li, L. On non-monotonicity height of piecewise monotone functions. Aequat. Math. 95, 401–414 (2021). https://doi.org/10.1007/s00010-021-00796-9

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