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ω-limit sets for maps of the interval

  • Louis Block (a1) and Ethan M. Coven (a2)

Abstract

Let f denote a continuous map of a compact interval to itself, P(f) the set of periodic points of f and Λ(f) the set of ω-limit points of f. Sarkovskǐi has shown that Λ(f) is closed, and hence ⊆Λ(f), and Nitecki has shown that if f is piecewise monotone, then Λ(f)=. We prove that if x∈Λ(f)−, then the set of ω-limit points of x is an infinite minimal set. This result provides the inspiration for the construction of a map f for which Λ(f)≠.

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Copyright

References

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[B]Birkhoff, G. D.. Dynamical Systems. Amer. Math. Soc. Colloq. Publ., vol. 9. Amer. Math. Soc: Providence, RI, 1927.
[C]Coppel, A.. Continuous maps on an interval. Xeroxed notes, 1984.
[CN]Coven, E. & Nitecki, Z.. Non-wandering sets of powers of maps of the interval. Ergod. Th. & Dynam. Sys. 1 (1981), 931.
[D]Delahaye, J. P.. Fonctions admettant des cycles d'order n'importe quelle puissance de 2 et aucun autre cycle. C.R. Acad. Sci. 291 (1980), A323–A325. Addendum A671.
[N1]Nitecki, Z.. Periodic and limit orbits and the depth of the center for piecewise monotone interval maps. Proc. Amer. Math. Soc. 80 (1980), 511514.
[N2Nitecki, Z.. Topological dynamics on the interval, Ergodic theory and dynamical systems II. (College Park, MD, 1979–80), pp. 1–73, Progress in Math. 21. Birkhauser: Boston, 1982.
[SI]Sarkovskˇi, A. N.. On a theorem of G. D. Birkhoff (Ukrainian, Russian and English summaries). Dopovidi Akad. Nauk Ukrain. RSR Ser. A, (1967), 429432.
[S2]Sarkovskˇi, A. N.. On some properties of discrete dynamical systems, Sur la theorie de l'iteration et ses applications. Colloque international du CNRS, No. 332, Toulouse, 1982.
[SK]Sarkovskˇi, A. N. & Kenzegulov, H. K.. On properties of the set of limit points of an iterative sequence of a continuous function (Russian). Voz. Mat. Sb. Vyp. 3 (1965), 343348.
[X]Xiong, J.-C.. Ω(f|Ω(f)) = for every continuous map f of the interval. Kexue Tongbao (English Ed.) 28 (1983), 2123.

ω-limit sets for maps of the interval

  • Louis Block (a1) and Ethan M. Coven (a2)

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