Abstract
We show that if \(f: \mathbb R\rightarrow \mathbb R\) is a continuous transitive map and \(\gamma :[0,1] \rightarrow \mathbb R\) is a nonconstant curve having finite total variation, then the total variation of \(f^n\circ \gamma \) tends to infinity exponentially as \(n \rightarrow \infty \). A similar result is also proved for a Devaney chaotic map on a graph G.
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Acknowledgements
We would like to thank the referees, whose comments and suggestions helped us to significantly improve the manuscript. The authors also wish to thank Sampad Lahiry for some useful discussions and they acknowledge the help of Dominik Kwietniak for some references and suggestions. The first author was supported partially by Department of Science and Technology (DST), Govt. of India, under the Scheme “Fund for Improvement of S &T Infrastructure” (FIST) [File No. SR/FST/MS-I/2019/41]. The research of second author was funded by UGC [NTA Ref. No. 201610319430], Govt. of India and the work of third author was funded by CSIR [file No. 08/155(0068)/2019-EMR-I], Govt. of India.
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The main problems and all the results are due to the above authors. The authors also wish to thank Sampad Lahiry for some useful discussions and they acknowledge the help of Dominik Kwietniak for some references and suggestions.
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Banerjee, K., Bhattacharyya, A. & Mondal, S. Total Variation of a Curve Under Chaos on the Real Line and on a Finite Graph. Qual. Theory Dyn. Syst. 23, 16 (2024). https://doi.org/10.1007/s12346-023-00871-3
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DOI: https://doi.org/10.1007/s12346-023-00871-3