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

Blokh, A. M.

doi:10.1090/S0002-9947-1992-1153009-7

The set of all iterates is nowhere dense in $C([0,1],[0,1])$
HTML articles powered by AMS MathViewer

by A. M. Blokh

Trans. Amer. Math. Soc. 333 (1992), 787-798

DOI: https://doi.org/10.1090/S0002-9947-1992-1153009-7

PDF | 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.

References

S. J. Agronsky, A. M. Bruckner, and M. Laczkovich, Dynamics of typical continuous functions, J. London Math. Soc. (2) 40 (1989), no. 2, 227–243. MR 1044271, DOI 10.1112/jlms/s2-40.2.227
Louis Block and Ethan M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc. 300 (1987), no. 1, 297–306. MR 871677, DOI 10.1090/S0002-9947-1987-0871677-X
Louis Block, John Guckenheimer, MichałMisiurewicz, and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. MR 591173
A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk 37 (1982), no. 2(224), 189–190 (Russian). MR 650765

The "spectral" decomposition for one-dimensional maps

Paul D. Humke and Miklós Laczkovich, Approximations of continuous functions by squares, Ergodic Theory Dynam. Systems 10 (1990), no. 2, 361–366. MR 1062763, DOI 10.1017/S0143385700005599
Paul D. Humke and M. Laczkovich, The Borel structure of iterates of continuous functions, Proc. Edinburgh Math. Soc. (2) 32 (1989), no. 3, 483–494. MR 1015490, DOI 10.1017/S0013091500004727
K. Simon, Typical continuous functions are not iterates, Acta Math. Hungar. 55 (1990), no. 1-2, 133–134. MR 1077067, DOI 10.1007/BF01951395
Károly Simon, The set of second iterates is nowhere dense in $C$, Proc. Amer. Math. Soc. 111 (1991), no. 4, 1141–1150. MR 1033961, DOI 10.1090/S0002-9939-1991-1033961-5
K. Simon, The iterates are not dense in $C$, Math. Pannon. 2 (1991), no. 1, 71–76. MR 1119725