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A counterexample in dynamical systems of the interval

Authors: Hsin Chu and Jin Cheng Xiong
Journal: Proc. Amer. Math. Soc. 97 (1986), 361-366
MSC: Primary 58F20; Secondary 58F08
MathSciNet review: 835899
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Abstract: In [1] it was proved that if the recurrent points of a continuous map of the unit interval form a closed set, then this map has no periodic point with period not equal to a power of 2, i.e. this map is of type $ {2^\infty }$. In this paper we will construct a continuous map of the interval which is of type $ {2^\infty }$ and for which the set of recurrent points is not closed. By such a counterexample it may be shown that some of the results announced in [2] are not correct.

References [Enhancements On Off] (What's this?)

  • [1] Xiong Jincheng, The periods of periodic points of continuous self-maps of the interval whose recurrent points form a closed set, J. China Univ. Sci. Tech. 13 (1983), 134-135. MR 701790 (84h:58124b)
  • [2] A. M. Bloh, The asymptotic behaviour of one-dimensional dynamical systems, Uspekhi Mat. Nauk 37 (1982), 175-176.
  • [3] L. Block, Stability of periodic orbits in the theorem of Šarkovski, Proc. Amer. Math. Soc. 81 (1981), 335-336. MR 593484 (82b:58071)
  • [4] A. Weil, Two lectures on number theory, past and present, Enseign. Math. 20 (1974), 87-110. MR 0366788 (51:3034)

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Keywords: Recurrent point, periodic point, type $ {2^\infty }$
Article copyright: © Copyright 1986 American Mathematical Society

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