Total Variation of a Curve Under Chaos on the Real Line and on a Finite Graph

• Kuntal Banerjee ^[1] ; Anubrato Bhattacharyya ^[1] ; Subhamoy Mondal ^[1]
  1. [1] Presidency University

     Presidency University

     India

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 1, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We show that if f : R → R is a continuous transitive map and γ : [0, 1] → R is a nonconstant curve having finite total variation, then the total variation of f n ◦ γ tends to infinity exponentially as n → ∞. A similar result is also proved for a Devaney chaotic map on a graph G

• Referencias bibliográficas
  • 1. Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc 114, 309–319 (1965)
  • 2. Alseda, L., Kolyada, S., Llibre, J., Snoha, L.: Entropy and periodic points for transitive maps. Trans. Am. Math. Soc 351(4), 1551–1573...
  • 3. Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Am. Math. Mon. 99(4), 332–334 (1992)
  • 4. Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge University Press, Cambridge (1998)
  • 5. Cánovas, J.S.: Topological entropy of continuous transitive maps on the real line. Dyn. Syst. 24(4), 473–483 (2009)
  • 6. Chen, G., Huang, T., Huang, Y.: Chaotic behaviour of interval maps and total variations of iterates. Int. J. Bifurc. Chaos 14(7), 2161–2186...
  • 7. Devaney, R.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Boston (1989)
  • 8. Gomes, J., Carneiro, M.: Polynomial entropy for interval maps and lap number. Qual. Theory Dyn. Syst. 20, 1–16 (2021)
  • 9. Hara ´nczyk, G., Kwietniak, D., Oprocha, P.: A note on transitivity, sensitivity and chaos for graph maps. J. Differ. Equ. Appl. 17(10),...
  • 10. Hara ´nczyk, G., Kwietniak, D., Oprocha, P.: Topological structure and entropy of mixing graph maps. Ergod. Theory Dyn. Syst. 34(5), 1587–1614...
  • 11. Huang, Y., Chen, G., Ma, D.: Rapid fluctuations of chaotic maps on RN . J. Math. Anal. Appl. 323(1), 228–252 (2006)
  • 12. Juang, J., Shieh, S.: On the total variation, topological entropy and sensitivity for interval maps. J. Math. Anal. Appl. 341(2), 1055–1067...
  • 13. Li, S.H.: ω-Chaos and topological entropy. Trans. Am. Math. Soc. 339, 243–249 (1993)
  • 14. Llibre, J., Misiurewicz, M.: Horseshoe, entropy and periods for graph maps. Topology 32(3), 649–664 (1993)
  • 15. Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings. Stud. Math. 67(1), 45–63 (1980)
  • 16. Miyazawa, M.: Chaos and entropy for circle maps. Tokyo J. Math. 25(2), 453–458 (2002)
  • 17. Miyazawa, M.: Chaos and entropy for graph maps. Tokyo J. Math. 27(1), 221–225 (2004)
  • 18. Ruette, S.: Chaos on the Interval. American Mathematical Society, Providence (2017)