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Polynomial Entropy of Induced Maps of Circle and Interval Homeomorphisms

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Abstract

We compute the polynomial entropy of the induced maps on hyperspace for a homeomorphism f of an interval or a circle with finitely many non-wandering points. Also, we give a generalization for the case of an interval homeomorphism with an infinite non-wandering set.

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References

  1. Acosta, G., Illanes, A., Méndez-Lango, H.: The transitivity of induced maps. Topology Appl. 156, 1013–1033 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arbieto, A., Bohorquez, J.: Shadowing, topological entropy and recurrence of induced Morse-Smale diffeomorphisms. arXiv:2203.13356 (2022)

  3. Artigue, A., Carrasco-Olivera, D., Monteverde, I.: Polynomial entropy and expansivity. Acta Math. Hungar. 152, 140–149 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Banks, J.: Chaos for induced hyperspace map. Chaos Solitons Fract. 25, 681–685 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauer, W., Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79(2), 81–92 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bernard, P., Labrousse, C.: An entropic characterization on the flat metric on the two torus. Geom. Dedicata 180, 187–201 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borsuk, K., Ulam, S.: On symmetric products of topological spaces. Bull Amer. Math. Soc. 37, 875–882 (1931)

    Article  MathSciNet  MATH  Google Scholar 

  8. Correa, J., de Paula, H.: Polynomial entropy of Morse–Smale diffeomorphisms on surfaces. arXiv:2203.08336v2 (2022)

  9. Daghar, A., Naghmouchi, I.: Entropy of induced maps of regular curves homeomorphisms. Chaos Solitons Fract. 157, 111988 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gomes, J.B., Carneiro, M.J.D.: Polynomial entropy for interval maps and lap number. Qual. Theory Dyn. Syst. 20(1), 1–16 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hauseux, L., Le Roux, F.: Entropie polynomiale des homémorphismes de Brouwer. Ann. Henri Lebesgue 2, 39–57 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hernández, P., Méndez, H.: Entropy of induced dendrite homeomorphisms. Topology Proc. 47, 191–205 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Katić, J., Perić, M.: On the polynomial entropy for Morse gradient systems. Math. Slovaca 69(3), 611–624 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kwietniak, D., Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces. Chaos Solitons Fract. 33(1), 76–86 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Labrousse, C.: Polynomial growth of the volume of balls for zero-entropy geodesic systems. Nonlinearity 25, 3049–3069 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Labrousse, C.: Flat metrics are strict local minimizers for the polynomial entropy. Regul. Chaotic Dyn. 17, 47–491 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Labrousse, C.: Polynomial entropy for the circle homeomorphisms and for \(C^1\) nonvanishing vector fields on \(\mathbb{T}^{2}\). arXiv:1311.0213 (2013)

  18. Labrousse, C., Marco, J.P.: Polynomial entropies for Bott integrable Hamiltonian systems. Regul. Chaotic Dyn. 19, 374–414 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lampart, M., Raith, P.: Topological entropy for set valued maps. Nonlinear Anal. 73(6), 1533–1537 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Marco, J.P.: Dynamical complexity and symplectic integrability. arXiv:0907.5363v1 (2009)

  21. Marco, J.P.: Polynomial entropies and integrable Hamiltonian systems. Regul. Chaotic Dyn. 18(6), 623–655 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Robinson, C.: Dynamical Systems. Studies in Advanced Mathematics, CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  23. Román-Flores, H.: A note on transitivity in set-valued discrete systems. Chaos Solitons Fract. 17(1), 99–104 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Roth, S., Roth, Z., Snoha, L’.: Rigidity and flexibility of polynomial entropy. arXiv:2107.13695v1 (2021)

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Acknowledgements

The authors thank the anonymous referees for many valuable comments and suggestions.

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Authors and Affiliations

  1. Matematički institut SANU, Knez Mihailova 36, Beograd, 11000, Serbia

    Maša Ɖorić

  2. Department of Mathematics, University of Belgrade Matematički fakultet, Studentski trg 16, Beograd, 11000, Serbia

    Jelena Katić

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  1. Maša Ɖorić
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  2. Jelena Katić
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Both authors contributed equally in the writing of the manuscript.

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Correspondence to Jelena Katić.

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This work is partially supported by Serbian Ministry of Education, Science and Technological Development through Faculty of Mathematics, Univeristy of Belgrade.

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Ɖorić, M., Katić, J. Polynomial Entropy of Induced Maps of Circle and Interval Homeomorphisms. Qual. Theory Dyn. Syst. 22, 103 (2023). https://doi.org/10.1007/s12346-023-00806-y

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