Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A ``universal'' dynamical system generated by a continuous map of the interval


Authors: David Pokluda and Jaroslav Smítal
Journal: Proc. Amer. Math. Soc. 128 (2000), 3047-3056
MSC (1991): Primary 58F12, 58F08, 58F03, 26A18
DOI: https://doi.org/10.1090/S0002-9939-00-05679-3
Published electronically: March 3, 2000
MathSciNet review: 1712885
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

In this paper we show that there is a continuous map $f:I\rightarrow I$of the interval such that any $\omega$-limit set $W$ of any continuous map $g:I\rightarrow I$ can be transformed by a homeomorphism $I\rightarrow I$ to an $\omega$-limit set $\tilde W$ of $f$. Consequently, any nowhere-dense compact set and any finite union of compact intervals is a homeomorphic copy of an $\omega$-limit set of $f$.


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

  • [1] S. J. Agronsky, A. M. Bruckner, J. G. Ceder and T. L. Pearson, The structure of $\omega$-limit sets for continuous functions, Real Anal. Exchange 15 (1989/90), 483 - 510. MR 91i:26008
  • [2] L. Block and W. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics 1513 (1991), Springer, New York, Heidelberg, Berlin. MR 93g:58091
  • [3] A. Blokh, A. M. Bruckner, P. D. Humke and J. Smítal, The space of $\omega$-limit sets of a continuous map of the interval, Trans. Amer. Math. Soc. 348 (1996), 1357 - 1372. MR 96j:58089
  • [4] A. M. Bruckner, Some problems concerning the iteration of real functions, Atti Sem. Mat. Fiz. Univ. Modena 41 (1993), 195 - 203. MR 94h:58114
  • [5] A. M. Bruckner and J. Smítal, The structure of $\omega$-limit sets for continuous maps of the interval, Math. Bohemica 117 (1992), 42 - 47. MR 93a:26002
  • [6] M. J. Evans, P. D. Humke, C. M. Lee, and R. O'Malley, Characterizations of turbulent one-dimensional mappings via $\omega$-limit sets, Trans. Amer. Math. Soc. 326 (1991), 261 - 280; correction, ibid. 333 (1992), 939 - 940. MR 91j:58133
  • [7] P. S. Keller, Chaotic behavior of Newton's method, Real Anal. Exchange 18 (1992/93), 490 - 507. MR 94j:26008
  • [8] A. N. Sharkovsky, Attracting and attracted sets, Soviet Math. Dokl. 6 (1965), 268 - 270.
  • [9] W. Sierpinski, Cardinal and Ordinal Numbers, Polish Scientific Publishers, Warsaw, 1958. MR 20:2288

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58F12, 58F08, 58F03, 26A18

Retrieve articles in all journals with MSC (1991): 58F12, 58F08, 58F03, 26A18


Additional Information

David Pokluda
Affiliation: Institute of Mathematics, Silesian University, 746 01 Opava, Czech Republic
Email: David.Pokluda@fpf.slu.cz

Jaroslav Smítal
Affiliation: Institute of Mathematics, Silesian University, 746 01 Opava, Czech Republic
Email: smital@fpf.slu.cz

DOI: https://doi.org/10.1090/S0002-9939-00-05679-3
Received by editor(s): November 1, 1998
Published electronically: March 3, 2000
Additional Notes: This research was supported, in part, by contract No. 201/97/0001 from the Grant Agency of the Czech Republic. Support of this institution is gratefully acknowledged.
Communicated by: Michael Handel
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society