Each nowhere dense nonvoid closed set in Rn is a σ-limit set

Andrei Sivak

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 2, page 183-190
  • ISSN: 0016-2736

Abstract

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We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in ℝ n , n ≥ 1, is a σ-limit set for some continuous map.

How to cite

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Sivak, Andrei. "Each nowhere dense nonvoid closed set in Rn is a σ-limit set." Fundamenta Mathematicae 149.2 (1996): 183-190. <http://eudml.org/doc/212116>.

@article{Sivak1996,
abstract = {We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in $ℝ^n$, n ≥ 1, is a σ-limit set for some continuous map.},
author = {Sivak, Andrei},
journal = {Fundamenta Mathematicae},
keywords = {dynamical systems; iterations; continuous maps},
language = {eng},
number = {2},
pages = {183-190},
title = {Each nowhere dense nonvoid closed set in Rn is a σ-limit set},
url = {http://eudml.org/doc/212116},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Sivak, Andrei
TI - Each nowhere dense nonvoid closed set in Rn is a σ-limit set
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 2
SP - 183
EP - 190
AB - We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in $ℝ^n$, n ≥ 1, is a σ-limit set for some continuous map.
LA - eng
KW - dynamical systems; iterations; continuous maps
UR - http://eudml.org/doc/212116
ER -

References

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  1. [1] S. Agronsky, A. Bruckner, J. Ceder and T. Pearson, The structure of ω-limit sets for continuous functions, Real Anal. Exchange 15 (1989-90), 483-510. Zbl0728.26006
  2. [2] A. Bruckner and J. Smítal, The structure of ω-limit sets for continuous maps of the interval, Math. Bohem. 117 (1992), 42-47. Zbl0762.26003
  3. [3] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, N.J., 1981. Zbl0459.28023
  4. [4] H. Hilmy, Sur les centres d'attraction minimaux dans les systèmes dynamiques, Compositio Math. 3 (1936), 227-238. Zbl62.0992.11
  5. [5] N. Kryloff et N. Bogoliouboff [N. Krylov et N. Bogolyubov], La théorie générale de la mesure dans son application à l'étude des systèmes de la mécanique non linéaire, Ann. of Math. (2) 38 (1937), 65-113. Zbl0016.08604
  6. [6] V. V. Nemytskiĭ and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton, N.J., 1960. Zbl0089.29502
  7. [7] A. N. Sharkovskiĭ, Attracting and attracted sets, Dokl. Akad. Nauk SSSR 160 (1965), 1036-1038 (in Russian); English transl.: Soviet Math. Dokl. 6 (1965), 268-270. 
  8. [8] A. G. Sivak, The structure of minimal attraction centers of trajectories of continuous maps of the interval, Real Anal. Exchange 20 (1994/95), 125-133. Zbl0824.54021

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