Abstract
We propose a new definition of preimage entropy dimension for continuous maps on compact metric spaces, investigate fundamental properties of the preimage entropy dimension, and compare the preimage entropy dimension with the topological entropy dimension. The defined preimage entropy dimension satisfies various basic properties of topological entropy dimension, for example, the preimage entropy dimension of a subsystem is bounded by that of the original system and topologically conjugated systems have the same preimage entropy dimension. Also, we discuss the relation between the preimage entropy dimension and the preimage entropy.
Funding Statement
The first author was supported by the Foundation in higher
education institutions of Henan Province, China (No. 19A110030); the National Natural Science
Foundation of China (No. 11401363); the Foundation for the Training of Young Key Teachers in
Colleges and Universities in Henan Province, China (No. 2018GGJS134). The second author was
supported by the National Natural Science Foundation of China (No. 11801193). The third author
was supported by the National Natural Science Foundation of China (No. 11971236); China
Postdoctoral Science Foundation (No. 2016M591873), and China Postdoctoral Science Special
Foundation (No. 2017T100384). The work was also funded by the Priority Academic Program
Development of Jiangsu Higher Education Institutions. We would like to express our gratitude
to Tianyuan Mathematical Center in Southwest China (11826102), Sichuan University and Southwest
Jiaotong University for their support and hospitality.
Acknowledgments
The authors are grateful to the anonymous referees for their comments, which helped to improve the paper.
Citation
Lei Liu. Xiaomin Zhou. Xiaoyao Zhou. "Preimage Entropy Dimension of Topological Dynamical Systems." Taiwanese J. Math. 25 (6) 1225 - 1239, December, 2021. https://doi.org/10.11650/tjm/210704
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