Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



A $C^1$ function for which the $\omega$-limit points are not contained in the closure of the periodic points

Authors: Emma D'Aniello and T. H. Steele
Journal: Trans. Amer. Math. Soc. 355 (2003), 2545-2556
MSC (2000): Primary 26A18; Secondary 54H20
Published electronically: February 4, 2003
MathSciNet review: 1974002
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • 1. L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Math., vol. 1513, Springer-Verlag, Berlin, 1992. MR 93g:58091
  • 2. L. Block and E. M. Coven, $\omega$-limit sets for maps of the interval, Ergodic Theory Dynamical Systems, vol. 6, pp. 335-344, 1986. MR 88a:58165
  • 3. Hsin Chu and J. Xiong, A counterexample in dynamical systems of the interval, Proc. Amer. Math. Soc., vol. 97, Number 2, pp. 361-366, 1986. MR 87i:58140
  • 4. E. M. Coven and E. D'Aniello, Chaos for maps of the interval via $\omega$-limit points and periodic points, Att. Sem. Mat. Fis. Univ. Modena, vol. 49, pp. 523-530, 2001. MR 2002j:37044
  • 5. E. M. Coven, J. Madden, and Z. Nitecki, A note on generic properties of continuous maps, Ergodic Theory Dynamical Systems II, Progress in Math., vol. 21, pp. 97-101, Birkhaüser, Boston, 1982. MR 84c:58068
  • 6. A. M. Bruckner and J. Ceder, Chaos in terms of the map $x \rightarrow \omega(x,f)$, Pacific J. Math., vol. 156, No. 1, pp. 63-96, 1992. MR 93g:58092
  • 7. R. L. Devaney, An introduction to chaotic dynamical systems, Benjamin/Cummings Publ. Co., 1986. MR 87e:58142
  • 8. V. V. Fedorenko, A. N. Sharkovskii, and J. Smital, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc., vol. 110, pp. 141-148, 1990. MR 91a:58148
  • 9. M. V. Jakobson, On smooth mappings of the circle into itself, Math. Sbornik, vol. 14, pp. 161-185, 1971.
  • 10. Z. Nitecki, Periodic and limit orbits and the depth of the center for piecewise monotone interval maps, Proc. Amer. Math. Soc., vol. 80, pp. 511-514, 1980. MR 81j:58068
  • 11. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math., vol. 89, pp. 1010-1021, 1967. MR 37:2257
  • 12. H. L. Royden, Real Analysis, Third Edition, Macmillan Publishing Company, New York, 1988. MR 90g:00004
  • 13. A. N. Sharkovskii, On a theorem of G. D. Birkhoff, Dopovidi Akad. Nauk Ukrain RSR Ser. A, pp. 429-432, 1967. MR 35:3646
  • 14. A. N. Sharkovskii, Attracting sets containing no cycles, Ukrainian Math. Z. 20, pp. 136-142 (Russian), 1968. MR 37:908
  • 15. B. Schweizer and J. Smital, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., vol. 344, no. 2, pp. 737-754, 1994. MR 94k:58091
  • 16. J. Smital, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., vol. 297, pp. 269-282, 1986. MR 87m:58107
  • 17. T. H. Steele, Iterative stability in the class of continuous functions, Real Analysis Exchange 24(2), pp. 765-780, 1998-99. MR 2001d:26008
  • 18. L. S. Young, A closing lemma on the interval, Inventiones Math., vol. 54, pp. 179-187, 1979. MR 80k:58084

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 26A18, 54H20

Retrieve articles in all journals with MSC (2000): 26A18, 54H20

Additional Information

Emma D'Aniello
Affiliation: Dipartimento di Matematica, Seconda Università degli Studi di Napoli, Via Vivaldi 43, 81100 Caserta, Italia

T. H. Steele
Affiliation: Department of Mathematics, Weber State University, Ogden, Utah 84408-1702

Received by editor(s): May 20, 2002
Received by editor(s) in revised form: August 13, 2002
Published electronically: February 4, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society