Iteration changes discontinuity into smoothness (Ⅱ): oscillating case

Tianqi Luo; Xiaohua Liu; Tianqi Luo; Xiaohua Liu

doi:10.3934/math.2023441

AIMS Mathematics

2023, Volume 8, Issue 4: 8793-8810. doi: 10.3934/math.2023441

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Research article

Iteration changes discontinuity into smoothness (Ⅱ): oscillating case

Tianqi Luo ,
Xiaohua Liu ^,

School of Mathematics and Physics, Leshan Normal University, Leshan 614000, China

Received: 15 April 2022 Revised: 26 June 2022 Accepted: 12 July 2022 Published: 08 February 2023
MSC : 37E05, 39B12

It has been shown that a self-mapping with exactly one removable or jumping discontinuity may have a $ C^1 $ smooth iterate of the second-order. However, some examples show that a self-mapping with exactly one oscillating discontinuity may also have a $ C^1 $ smooth iterate of the second-order, indicating that iteration can turn a self-mapping with exactly one oscillating discontinuity into a $ C^1 $ smooth one. In this paper, we study piecewise $ C^1 $ self-mappings on the open interval $ (0, 1) $ having only one oscillating discontinuity. We give necessary and sufficient conditions for those self-mappings whose second-order iterates are $ C^1 $ smooth.
- iteration,
- oscillating discontinuity,
- $ C^{1} $ smooth,
- piecewise $ C^{1} $ smooth
Citation: Tianqi Luo, Xiaohua Liu. Iteration changes discontinuity into smoothness (Ⅱ): oscillating case[J]. AIMS Mathematics, 2023, 8(4): 8793-8810. doi: 10.3934/math.2023441

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Abstract

It has been shown that a self-mapping with exactly one removable or jumping discontinuity may have a $ C^1 $ smooth iterate of the second-order. However, some examples show that a self-mapping with exactly one oscillating discontinuity may also have a $ C^1 $ smooth iterate of the second-order, indicating that iteration can turn a self-mapping with exactly one oscillating discontinuity into a $ C^1 $ smooth one. In this paper, we study piecewise $ C^1 $ self-mappings on the open interval $ (0, 1) $ having only one oscillating discontinuity. We give necessary and sufficient conditions for those self-mappings whose second-order iterates are $ C^1 $ smooth.

References

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