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Dynamics of the iteration operator on the space of continuous self-maps


Authors: Murugan Veerapazham, Chaitanya Gopalakrishna and Weinian Zhang
Journal: Proc. Amer. Math. Soc. 149 (2021), 217-229
MSC (2010): Primary 39B12, 47H30; Secondary 37C25, 54H20
DOI: https://doi.org/10.1090/proc/15178
Published electronically: October 16, 2020
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Abstract: The semi-dynamical system of a continuous self-map is generated by iteration of the map, however, the iteration itself, being an operator on the space of continuous self-maps, may generate interesting dynamical behaviors. In this paper we prove that the iteration operator is continuous on the space of all continuous self-maps of a compact metric space and therefore induces a semi-dynamical system on the space. Furthermore, we characterize its fixed points and periodic points in the case that the compact metric space is a compact interval by discussing the Babbage equation. We prove that all orbits of the iteration operator are bounded but most fixed points are not stable. On the other hand, we prove that the iteration operator is not chaotic.


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Additional Information

Murugan Veerapazham
Affiliation: Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka Surathkal, Mangalore- 575 025, India
Email: murugan@nitk.edu.in

Chaitanya Gopalakrishna
Affiliation: Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka Surathkal, Mangalore- 575 025, India
Email: cberbalaje@gmail.com

Weinian Zhang
Affiliation: Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China
Email: matzwn@126.com

DOI: https://doi.org/10.1090/proc/15178
Keywords: Iteration operator, periodic points, Babbage equation, topological transitivity, chaos.
Received by editor(s): December 23, 2019
Received by editor(s) in revised form: April 29, 2020
Published electronically: October 16, 2020
Additional Notes: The first author was supported by SERB, DST, Government of India, through the project $ECR/2017/000765$.
The third author is the corresponding author and was supported by NSFC # 11521061, # 11771307 and # 11831012 and by China MOE PCSIRT $IRT_{15R53}$.
Communicated by: Wenxian Shen
Article copyright: © Copyright 2020 American Mathematical Society