A topological invariant for pairs of maps

• Autorzy: Marcelo Polezzi , Claudemir Aniz
• Języki publikacji: EN

Abstrakty

EN

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝn, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.

Słowa kluczowe

EN

Contact orders for pairs of maps; Sharkovsky’s theorem; discrete dynamical systems; 26A18; 37E05

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 2
• Strony: 294-303

Daty

wydano

2006-06-01

online

2006-06-01

Twórcy

autor

Marcelo Polezzi

• Universidade Estadual de Mato Grosso do Sul-(UEMS)

autor

Claudemir Aniz

• Universidade Federal de Mato Grosso do Sul-(UFMS)

Bibliografia

• [1] R. Barton and K. Burns: “A Simple Special Case of Sharkovskii’s Theorem”, Amer. Math. Monthly, Vol. 107(10), (2000), pp. 932–933.
• [2] N. Bhatia: “New Proof and Extension of Sarkovskii’s Theorem”, Far East J. Math. Sci., Special Volume, Part I, (1996), pp. 53–68.
• [3] B.-S. Du: “A Simple Proof of Sharkovsky’s Theorem”, Amer. Math. Monthly, Vol. 111(7), (2004), pp. 595–599.
• [4] S. Elayadi: “On a Converse of Sharkovsky’s Theorem”, Amer. Math. Monthly, Vol. 103, (1996), pp. 386–392.
• [5] V. Kannan, P.V.S.P. Saradhi and S.P. Seshasai: “A Generalization of Sarkovskii’s Theorem to Higher Dimensions”, J. Nat. Acad. Math. India, Vol. 11, (1997), pp. 69–82.
• [6] T.-Y. Li and J.A. Yorke: “Period Three Impies Chaos”, Amer. Math. Monthly, Vol. 82(10), (1975), pp. 985–992.
• [7] V.J. López and L. Snoha: “All Maps of Type 2∞ are Boundary Maps”, Proc. Amer. Math. Soc., Vol. 125(6), (1997), pp. 1667–1673. http://dx.doi.org/10.1090/S0002-9939-97-03452-7
• [8] M. Polezzi and C. Aniz: “A Šarkovskii-Type Theorem for Pairs of Maps”, Far East J. Dynamical Systems, Vol. 7(1), (2005), pp. 65–75.
• [9] A.N. Sharkovsky: “Coexistence of cycles of a continuous map of a line into itself”, Ukrain. Mat. Zh., Vol. 16(1), (1964), pp. 61–71 (Russian); Internat. J. Bifur. Chaos Appl. Sci. Engrg., Vol. 5, (1995), pp. 1263–1273 (English).
• [10] A.N. Sharkovsky: “On cycles and the structure of a continuous map”, Ukrain. Mat. Zh., Vol. 17(3), (1965), pp. 104–111 (Russian).
• [11] P.D. Straffin, Jr.: “Periodic Points of Continuous Functions”, Math. Mag., Vol. 51(2), (1978), pp. 99–105.

Identyfikatory

DOI

10.2478/s11533-006-0009-6