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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The set of all iterates is nowhere dense in $C([0,1],[0,1])$
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by A. M. Blokh PDF
Trans. Amer. Math. Soc. 333 (1992), 787-798 Request permission

Abstract:

We prove that if a mixing map $f:[0,1] \to [0,1]$ belongs to the ${C^0}$-closure of the set of iterates and $f(0) \ne 0$, $f(1) \ne 1$ then $f$ is an iterate itself. Together with some one-dimensional techniques it implies that the set of all iterates is nowhere dense in $C([0,1],[0,1])$ giving the final answer to the question of A. Bruckner, P. Humke and M. Laczkovich.
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 333 (1992), 787-798
  • MSC: Primary 26A18; Secondary 58F08
  • DOI: https://doi.org/10.1090/S0002-9947-1992-1153009-7
  • MathSciNet review: 1153009