A 𝐶¹ function for which the 𝜔-limit points are not contained in the closure of the periodic points

D’Aniello, Emma; Steele, T.

doi:10.1090/S0002-9947-03-03258-6

A $C^1$ function for which the $\omega$-limit points are not contained in the closure of the periodic points
HTML articles powered by AMS MathViewer

by Emma D’Aniello and T. H. Steele

Trans. Amer. Math. Soc. 355 (2003), 2545-2556

DOI: https://doi.org/10.1090/S0002-9947-03-03258-6

Published electronically: February 4, 2003

PDF | Request permission

Abstract:

We develop a $C^1$ function $f: [- \frac {1}{6}, 1] \rightarrow [- \frac {1}{6}, 1]$ for which $\Lambda (f) \not = \overline {P(f)}$. This answers a query from Block and Coppel (1992).

References

L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513, DOI 10.1007/BFb0084762
Louis Block and Ethan M. Coven, $\omega$-limit sets for maps of the interval, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 335–344. MR 863198, DOI 10.1017/S0143385700003539
Hsin Chu and Jin Cheng Xiong, A counterexample in dynamical systems of the interval, Proc. Amer. Math. Soc. 97 (1986), no. 2, 361–366. MR 835899, DOI 10.1090/S0002-9939-1986-0835899-0
Ethan M. Coven and Emma D’Aniello, Chaos for maps of the interval via $\omega$-limit points and periodic points, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), no. 2, 523–530. MR 1878254
E. M. Coven, J. Madden, and Z. Nitecki, A note on generic properties of continuous maps, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 97–101. MR 670076
A. M. Bruckner and J. Ceder, Chaos in terms of the map $x\to \omega (x,f)$, Pacific J. Math. 156 (1992), no. 1, 63–96. MR 1182256
Robert L. Devaney, An introduction to chaotic dynamical systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986. MR 811850
V. V. Fedorenko, A. N. Šarkovskii, and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc. 110 (1990), no. 1, 141–148. MR 1017846, DOI 10.1090/S0002-9939-1990-1017846-5
M. V. Jakobson, On smooth mappings of the circle into itself, Math. Sbornik, vol. 14, pp. 161-185, 1971.
Zbigniew Nitecki, Periodic and limit orbits and the depth of the center for piecewise monotone interval maps, Proc. Amer. Math. Soc. 80 (1980), no. 3, 511–514. MR 581016, DOI 10.1090/S0002-9939-1980-0581016-9
Charles C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021. MR 226670, DOI 10.2307/2373414
H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. MR 1013117
O. M. Šarkovs′kiĭ, On a theorem of G. D. Birkhoff, Dopovīdī Akad. Nauk Ukraïn. RSR Ser. A 1967 (1967), 429–432 (Ukrainian, with Russian and English summaries). MR 0212781
A. N. Šarkovskiĭ, Attracting sets containing no cycles, Ukrain. Mat. Ž. 20 (1968), no. 1, 136–142 (Russian). MR 0225314
B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), no. 2, 737–754. MR 1227094, DOI 10.1090/S0002-9947-1994-1227094-X
J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269–282. MR 849479, DOI 10.1090/S0002-9947-1986-0849479-9
T. H. Steele, Iterative stability in the class of continuous functions, Real Anal. Exchange 24 (1998/99), no. 2, 765–780. MR 1704748, DOI 10.2307/44152994
Lai Sang Young, A closing lemma on the interval, Invent. Math. 54 (1979), no. 2, 179–187. MR 550182, DOI 10.1007/BF01408935