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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.

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References

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Branner, B., Hubbard, J.H.: The iteration of cubic polynomials I. Acta Math. (Djursholm) 160, 143–206 (1988/1992)

    Google Scholar 

  4. Branner, B., Hubbard, J.H.: The iteration of cubic polynomials II. Acta Math. (Djursholm) 169, 229-325 (1988/1992)

    Google Scholar 

  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. Jarczyk, W., Zhang, W.: Also multifunctions do not like iterative roots. Elemente Math. 62(2), 73–80 (2007)

    Google Scholar 

  7. Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)

  8. Lesmoir-Gordon, N., Rood, W., Edney, R.: Introducing Fractal Geometry. Icon Books, Cambridge (2006)

  9. Lesmoir-Gordon, N.: The Colours of Infinity: The Beauty and Power of Fractals. Springer, London (2010)

  10. Mandelbrot, B.B.: Fractals and Chaos: The Mandelbrot Set and Beyond. Springer, New York (2004)

  11. Powierza T.: Set-valued iterative square roots of bijections. Bull. Pol. Acad. Math. 47, 377–383 (1999)

    MATH  MathSciNet  Google Scholar 

  12. Powierza, T.: On functions with weak iterative roots. Aequationes Math. 63, 103–109 (2002)

    Google Scholar 

  13. Sun, D.: Iteration of quasi-polynomial of degree two (in Chinese). J. Math 24, 237–240 (2004)

    Google Scholar 

  14. Targonski, G.: Topics in Iteration Theory. Vandenhoeck and Ruprecht, Götingen (1981)

  15. Wu Z., Sun D.: The iteration of quasi-polynimials mappings (in Chinese). Acta. Math. Sci. A 26, 493–497 (2006)

    MATH  MathSciNet  Google Scholar 

  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. Yu Z., Yang L., Zhang W.: Discussion on polynomials having polynomial iterative roots. J. Symb. Comput. 47(10), 1154–1162 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Leshan Normal College, Leshan, Sichuan, 614004, People’s Republic of China

    Xiaohua Liu

  2. Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Haidian District, Beijing, 100190, People’s Republic of China

    Liu Liu

  3. Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, People’s Republic of China

    Weinian Zhang

Authors
  1. Xiaohua Liu
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  2. Liu Liu
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  3. Weinian Zhang
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Correspondence to Liu Liu.

Additional information

Supported by Scientific Research Fund of Sichuan Provincial Education Department under Grant 13ZA0106, SRFDP#20120181110062 and MOE IRT1273.

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Liu, X., Liu, L. & Zhang, W. Discontinuous function with continuous second iterate. Aequat. Math. 88, 243–266 (2014). https://doi.org/10.1007/s00010-013-0220-z

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  • DOI: https://doi.org/10.1007/s00010-013-0220-z

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