Aequationes mathematicae

, Volume 88, Issue 3, pp 243–266 | Cite as

Discontinuous function with continuous second iterate

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Abstract

It is easy to find some discontinuous functions whose iterates are continuous ones, showing that iteration can change discontinuity into continuity. In order to investigate this change and avoid complicated computation, in this paper we classify self-mappings on the compact interval [0, 1] each of which has only one discontinuous point. We give sufficient and necessary conditions under which the second order iterates are continuous functions.

Mathematics Subject Classification (2010)

39B12 37E05 

Keywords

Iteration continuity discontinuous point 
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References

  1. 1.
    Baker, I.N.: The iteration of polynomials and transcendental entire functions. J. Austral. Math. Soc. Ser. A. 30(4), 483–495 (1980/1981)Google Scholar
  2. 2.
    Bhattacharyya P., Arumaraj Y.E.: On the iteration of polynomials of degree 4 with real coeffcients. Ann. Acad. Sci. Fenn. Math. 6(2), 197–203 (1981)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Branner, B., Hubbard, J.H.: The iteration of cubic polynomials I. Acta Math. (Djursholm) 160, 143–206 (1988/1992)Google Scholar
  4. 4.
    Branner, B., Hubbard, J.H.: The iteration of cubic polynomials II. Acta Math. (Djursholm) 169, 229-325 (1988/1992)Google Scholar
  5. 5.
    Jarczyk, W., Powierza, T.: On the smallest set-valued iterative roots of bijections. Int. J. Bifur. Chaos 13, 1889–1893 (2003)Google Scholar
  6. 6.
    Jarczyk, W., Zhang, W.: Also multifunctions do not like iterative roots. Elemente Math. 62(2), 73–80 (2007)Google Scholar
  7. 7.
    Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)Google Scholar
  8. 8.
    Lesmoir-Gordon, N., Rood, W., Edney, R.: Introducing Fractal Geometry. Icon Books, Cambridge (2006)Google Scholar
  9. 9.
    Lesmoir-Gordon, N.: The Colours of Infinity: The Beauty and Power of Fractals. Springer, London (2010)Google Scholar
  10. 10.
    Mandelbrot, B.B.: Fractals and Chaos: The Mandelbrot Set and Beyond. Springer, New York (2004)Google Scholar
  11. 11.
    Powierza T.: Set-valued iterative square roots of bijections. Bull. Pol. Acad. Math. 47, 377–383 (1999)MATHMathSciNetGoogle Scholar
  12. 12.
    Powierza, T.: On functions with weak iterative roots. Aequationes Math. 63, 103–109 (2002)Google Scholar
  13. 13.
    Sun, D.: Iteration of quasi-polynomial of degree two (in Chinese). J. Math 24, 237–240 (2004)Google Scholar
  14. 14.
    Targonski, G.: Topics in Iteration Theory. Vandenhoeck and Ruprecht, Götingen (1981)Google Scholar
  15. 15.
    Wu Z., Sun D.: The iteration of quasi-polynimials mappings (in Chinese). Acta. Math. Sci. A 26, 493–497 (2006)MATHMathSciNetGoogle Scholar
  16. 16.
    Xu, L., Xu, S.: On iteration of linear fractional function and applications (in Chinese). Math. Pract. Theory 36, 225–228 (2006)Google Scholar
  17. 17.
    Yu Z., Yang L., Zhang W.: Discussion on polynomials having polynomial iterative roots. J. Symb. Comput. 47(10), 1154–1162 (2012)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsLeshan Normal CollegeLeshanPeople’s Republic of China
  2. 2.Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.Yangtze Center of Mathematics and Department of MathematicsSichuan UniversityChengduPeople’s Republic of China

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