On N-Distal Homeomorphisms

• E. Rego ^[2] ; J. C. Salcedo ^[1]
  1. [1] Universidade Federal do Rio de Janeiro

     Universidade Federal do Rio de Janeiro

     Brasil

     
  2. [2] Southern University of Science and Technology (China)
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 4, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this work we study N-distal properties for homeomorphisms on compact metric spaces. For instance, we define N-equicontinuity and prove that every N-equicontinuous system is N-distal. We introduce the notions of N-distal extensions and N-distal factors. We also prove that M-distal extensions of N-distal homeomorphisms are MN-distal and present a non-trivial N-distal factor for N-distal homeomorphism having Ellis semigroup with a unique minimal ideal. It is also shown that transitive N-distal homeomorphisms have at most N−1 minimal proper subsystems. Finally, we prove that topological entropy vanishes for countably-distal systems on compact metric spaces. These results generalize previous ones for distal systems (Fürstenberg in Am J Math 85:477–515, 1963; Parry in Zero entropy of distal and related transformations, Benjamin, New York, pp 383–389, 1967).

• Referencias bibliográficas
  • 1. Aoki, N., Hiraide, K.: Topological Theory of Dynamical Systems. North Holland, New York (1994)
  • 2. Aponte, J., Carrasco, D., Keonhee, L., Morales, C.A.: Some generalizations of distality. Topol. Methods Nonlinear Anal. 55, 533–552 (2020)
  • 3. Auslander, J.: On the proximal relation in topological dynamics. Proc. Am. Math. Soc. 11, 890–895 (1960)
  • 4. Auslander, J.: Minimal flows and their extensions., North-Holland Mathematics Studies, 153. Notas de Matemática [Mathematical Notes], 122....
  • 5. Blanchard, F., Glasner, E., Kolyada, S., Maass, A.: On Li–orke pairs. J. Reine Angew. Math. 547, 51–68 (2002)
  • 6. Bourbaki, N.: General Topology, vol. II. Springer, Berlin (1998)
  • 7. Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge University Press, Cambridge (2002)
  • 8. Ellis, D., Ellis, R.: Automorphisms and Equivalence Relations in Topological Dynamics. Cambridge University Press, Cambridge (2014)
  • 9. Ellis, R.: Distal transformation groups. Pacific J. Math. 8, 401–405 (1958)
  • 10. Ellis, R.: A semigroup associated with a transformation group. Trans. Am. Math. Soc. 94, 272–281 (1960)
  • 11. Ellis, R.: Lectures on Topological Dynamics. Benjamin, New York (1969)
  • 12. Ellis, R., Gottschalk, W.H.: Homomorphisms of transformation groups. Trans. Am. Math. Soc. 94, 258–271 (1960)
  • 13. Fürstenberg, H.: The structure of distal flows. Am. J. Math. 85, 477–515 (1963)
  • 14. Fürstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981)
  • 15. Kato, H.: Continuum-wise expansive homeomorphisms. Canad. J. Math. 45(3), 576–598 (1993)
  • 16. Keynes, H.B.: On the proximal relation being closed. Proc. Am. Math. Soc. 18, 518–522 (1967)
  • 17. Lee, K., Morales, C.A.: Distal points for Borel measures. Topol. Appl. 221, 524–533 (2017)
  • 18. Lindenstrauss, E.: Measurable distal and topological distal systems. Ergodic Theory Dyn. Syst. 19(4), 1063–1076 (1999)
  • 19. Metzger, R., Morales, C., Villavicencio, H.: Generalized Archimedean spaces and expansivity. Topol. Appl. 302, 1–8 (2021)
  • 20. Morales, C.: A generalization of expansivity. Discrete Contin. Dyn. Syst. 32(1), 293–301 (2012)
  • 21. Morales, C.A., Sirvent, Víctor F.: Expansive measures. IMPA Série D: Rio de Janeiro, Brazil (2013)
  • 22. Ornstein, D., Weiss, B.: Mean distality and tightness, Topology and its Applications 221, 524–533 (2017). Tr. Mat. Inst. Steklova 244...
  • 23. Parry, W.: Zero entropy of distal and related transformations, Topological Dynamics (Symposium, Colorado State Univ., Ft. Collins, Colo.,:...
  • 24. Tom Dieck, T.: Algebraic topology. EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich (2008)
  • 25. Utz, W.R.: Unstable homeomorphisms. Proc. Am. Math. Soc. 1, 769–774 (1950)
  • 26. Veech, W.A.: Point-distal flows. Am. J. Math. 92, 205–242 (1970)
  • 27. Xiong, J.: Chaos in a topologically transitive system Science in China Ser. A Math. 48(7), 929–939 (2005)
  • 28. Zimmer, R.J.: Ergodic actions with generalized discrete spectrum. Illinois J. Math. 20(4), 555–588 (1976)
  • 29. Zippin, L.: Transformation groups, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., pp. 191–221 (1941)