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Iterative roots of two-dimensional mappings

Part of: Low-dimensional dynamical systems

Published online by Cambridge University Press:  05 April 2023

Zhiheng Yu ,
Lin Li  and
Janusz Matkowski
Zhiheng Yu
Affiliation:
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China (yuzhiheng9@163.com)
Lin Li
Affiliation:
Department of Mathematics, Jiaxing University, Jiaxing, Zhejiang 314001, China (matlinl@zjxu.edu.cn)
Janusz Matkowski
Affiliation:
Institute of Mathematics, University of Zielona Góra, Szafrana 4a, Zielona Góra PL 65-516, Poland (J.Matkowski@wmie.uz.zgora.pl)
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Abstract

As a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.

Keywords

iterative roottwo-dimensional mappingmonotonicitypartial orderdifferentiability

MSC classification

Primary: 39B12: Iteration theory, iterative and composite equations
Secondary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 1 , February 2023 , pp. 241 - 258
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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