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

Published electronically:
March 3, 2000

MathSciNet review:
1712885

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

In this paper we show that there is a continuous map of the interval such that any -limit set of any continuous map can be transformed by a homeomorphism to an -limit set of . Consequently, any nowhere-dense compact set and any finite union of compact intervals is a homeomorphic copy of an -limit set of .

**[1]**S. J. Agronsky, A. M. Bruckner, J. G. Ceder, and T. L. Pearson,*The structure of 𝜔-limit sets for continuous functions*, Real Anal. Exchange**15**(1989/90), no. 2, 483–510. MR**1059418****[2]**L. S. Block and W. A. Coppel,*Dynamics in one dimension*, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR**1176513****[3]**Alexander Blokh, A. M. Bruckner, P. D. Humke, and J. Smítal,*The space of 𝜔-limit sets of a continuous map of the interval*, Trans. Amer. Math. Soc.**348**(1996), no. 4, 1357–1372. MR**1348857**, 10.1090/S0002-9947-96-01600-5**[4]**A. M. Bruckner,*Some problems concerning the iteration of real functions*, Atti Sem. Mat. Fis. Univ. Modena**41**(1993), no. 1, 195–203. MR**1225682****[5]**Andrew M. Bruckner and Jaroslav Smítal,*The structure of 𝜔-limit sets for continuous maps of the interval*, Math. Bohem.**117**(1992), no. 1, 42–47 (English, with Czech summary). MR**1154053****[6]**Michael J. Evans, Paul D. Humke, Cheng Ming Lee, and Richard J. O’Malley,*Characterizations of turbulent one-dimensional mappings via 𝜔-limit sets*, Trans. Amer. Math. Soc.**326**(1991), no. 1, 261–280. MR**1010884**, 10.1090/S0002-9947-1991-1010884-3**[7]**Paul S. Keller,*Chaotic behavior of Newton’s method*, Real Anal. Exchange**18**(1992/93), no. 2, 490–507. MR**1228417****[8]**A. N. Sharkovsky,*Attracting and attracted sets*, Soviet Math. Dokl.**6 (1965)**, 268 - 270.**[9]**Wacław Sierpiński,*Cardinal and ordinal numbers*, Polska Akademia Nauk, Monografie Matematyczne. Tom 34, Państwowe Wydawnictwo Naukowe, Warsaw, 1958. MR**0095787**

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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