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The set of all iterates is nowhere dense in $ C([0,1],[0,1])$


Author: A. M. Blokh
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
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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

DOI: https://doi.org/10.1090/S0002-9947-1992-1153009-7
Keywords: Iterates of maps, mixing maps, periodic points
Article copyright: © Copyright 1992 American Mathematical Society

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