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Dynamics of Baire-1 functions on the interval

  • Timothy H. Steele
Research Article
  • 15 Downloads

Abstract

Let Open image in new window with \(bB_{1}\) the set of Baire-1 self-maps of I. For Open image in new window , let Open image in new window be the collection of \(\omega \)-limit sets generated by f, and \(\mathrm{\Lambda } (f)=\bigcup _{x\in I}\omega (x,f)\) be the set of \(\omega \)-limit points of f. There exists S a residual subset of \(bB_{1}\) such that for any Open image in new window , the following hold:
  • For any \(x\in I\), the \(\omega \)-limit set \(\omega (x,f)\) is contained in the set of points at which f is continuous, and \(\omega (x,f)\) is an \( \infty \)-adic adding machine.

  • For any \(\varepsilon >0\), there exists a natural number M such that \( f^{m}(I)\subset B_{\varepsilon }(\mathrm{\Lambda } (f))\) whenever \(m>M\). Moreover, \( f:\mathrm{\Lambda } (f)\rightarrow \mathrm{\Lambda } (f)\) is a bijection, and \(\mathrm{\Lambda } (f)\) is closed.

  • The Hausdorff s-dimensional measure of \(\mathrm{\Lambda } (f)\) is zero for all \(s>0\).

  • The collection of \(\omega \)-limit sets \(\mathrm{\Omega } (f)\) is closed in the Hausdorff metric space.

  • If x is a point at which f is continuous, then (xf) is a point at which the map Open image in new window given by \( (x,f)\mapsto \omega (x,f)\) is continuous.

  • The n-fold iterate \(f^{n}\) is an element of \(bB_{1}\) for all natural numbers n.

  • The function f is non-chaotic in the senses of Devaney and Li–Yorke.

Keywords

Baire-1 function \(\omega \)-Limit set Odometer Generic 

Mathematics Subject Classification

54H20 37B99 26A18 

References

  1. 1.
    Agronsky, S.J., Bruckner, A.M., Laczkovich, M.: Dynamics of typical continuous functions. J. London Math. Soc. 40(2), 227–243 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Amer. Math. Monthly 99(4), 332–334 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bernardes Jr., N.C., Darji, U.B.: Graph theoretic structure of maps of the Cantor space. Adv. Math. 231(3–4), 1655–1680 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Block, L.S., Coppel, W.A.: Dynamics in One Dimension. Lecture Notes in Mathematics, vol. 1513. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  5. 5.
    Block, L., Keesling, J.: A characterization of adding machines. Topology Appl. 140(2–3), 151–161 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blokh, A.M.: The “spectral” decomposition for one-dimensional maps. In: Jones, C.K., Kirchgraber, U., Walther, H.-O. (eds.) Dynamics Reported: Expositions in Dynamical Systems (New Series), vol. 4, pp. 1–59. Springer, Berlin (1995)CrossRefGoogle Scholar
  7. 7.
    Blokh, A., Bruckner, A.M., Humke, P.D., Smítal, J.: The space of \(\omega \)-limit sets of a continuous map of the interval. Trans. Amer. Math. Soc. 348(4), 1357–1372 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bruckner, A.M., Bruckner, J.B., Thomson, B.S.: Real Analysis. Prentice-Hall, Upper Saddle River (1997)zbMATHGoogle Scholar
  9. 9.
    Bruckner, A.M., Ceder, J.: Chaos in terms of the map \(x\rightarrow \omega (x, f)\). Pacific J. Math. 156, 63–96 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bruckner, A.M., Petruska, G.: Some typical results on bounded Baire 1 functions. Acta Math. Hungar. 43(3–4), 325–333 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bruckner, A.M., Steele, T.H.: The Lipschitz structure of continuous self-maps of generic compact sets. J. Math. Anal. Appl. 118(3), 798–808 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Buescu, J., Stewart, I.: Liapunov stability and adding machines. Ergodic Theory Dynam. Systems 15(2), 271–290 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    D’Aniello, E., Darji, U.B., Steele, T.H.: Ubiquity of odometers in topological dynamical systems. Topology Appl. 156(2), 240–245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fedorenko, V.V., Šarkovskii, A.N., Smítal, J.: Characterizations of weakly chaotic maps of the interval. Proc. Amer. Math. Soc. 110(1), 141–148 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lehning, H.: Dynamics of typical continuous functions. Proc. Amer. Math. Soc. 123(6), 1703–1707 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nitecki, Z.: Topological dynamics on the interval. In: Katok, A. (ed.) Ergodic Theory and Dynamical Systems, vol. II. Progress in Mathematcis, vol. 21, pp. 1–73. Birkhäuser, Boston (1982)Google Scholar
  17. 17.
    Smítal, J.: Chaotic functions with zero topological entropy. Trans. Amer. Math. Soc. 297(1), 269–282 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Steele, T.H.: Continuity and chaos in discrete dynamical systems. Aequationes Math. 71(3), 300–310 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Steele, T.H.: Dynamics of typical Baire-1 functions on the interval. J. Appl. Anal. 23(2), 59–64 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Steele, T.H.: The space of \(\omega \)-limit sets for Baire-1 functions on the interval (submitted)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsWeber State UniversityOgdenUSA

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