Real Analysis Exchange

Towards a characterization of ω-limit sets for Lipschitz functions

T. H. Steele

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Abstract

Recent research has shown that there is a significant cleavage between the structure of ω sets for continuous functions, and the structure of ω-limit sets for Lipschitz functions. While every non-empty nowhere dense compact set is an ω-limit set for a continuous function, most of these sets cannot be an attractor for a Lipschitz function. When one considers the topological structure of these two classes of ω-limit sets, however, there is no such divergence. In this paper we investigate the necessarily measure based structural differences between these two classes of sets. We then use these results to work towards a characterization of ω-limit sets for Lipschitz functions.

Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 201-212.

Dates
First available in Project Euclid: 1 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1338515215

Mathematical Reviews number (MathSciNet)
MR1433608

Subjects
Primary: 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25]

Keywords
ω limit set Lipschitz function

Citation

Steele, T. H. Towards a characterization of ω-limit sets for Lipschitz functions. Real Anal. Exchange 22 (1996), no. 1, 201--212. https://projecteuclid.org/euclid.rae/1338515215


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References

  • A. Blokh, A. M. Bruckner, P. D. Humke and J. Smítal, The space of \w sets of a continuous map of the interval, MSRI Preprint, 063-94, 1994, 1–19.
  • A. M. Bruckner and J. G. Ceder, Chaos in terms of the map $x\mapsto\omega(x,f)$, Pac. J. Math., 156 (1992), 63–96.
  • A. M. Bruckner and J. Smítal, The structure of $\omega$-limit sets for continuous maps of the interval, Math. Bohemica, 117 (1992), 42–47.
  • A. M. Bruckner and J. Smítal, A characterization of $\omega$-limit sets of maps of the interval with zero topological entropy, Ergod. Th. and Dynam. Sys., 13 (1993), 7–19.
  • A. M. Bruckner and T. H. Steele, The Lipschitz structure of continuous self-maps of generic compact sets, J. Math. Anal. Appl., 188 (1994), 798–808.
  • J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269–282.
  • T. H. Steele, The topological structure of attractors for differentiable functions, Real Anal. Ex., 21 (1995), 181–193.