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.
@article{bwmeta1.element.doi-10_2478_s11533-006-0009-6,
author = {Marcelo Polezzi and Claudemir Aniz},
title = {A topological invariant for pairs of maps},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {294-303},
zbl = {1103.37027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0009-6}
}
Marcelo Polezzi; Claudemir Aniz. A topological invariant for pairs of maps. Open Mathematics, Tome 4 (2006) pp. 294-303. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0009-6/
[1] R. Barton and K. Burns: “A Simple Special Case of Sharkovskii’s Theorem”, Amer. Math. Monthly, Vol. 107(10), (2000), pp. 932–933. | Zbl 0979.37016
[2] N. Bhatia: “New Proof and Extension of Sarkovskii’s Theorem”, Far East J. Math. Sci., Special Volume, Part I, (1996), pp. 53–68. | Zbl 0928.54038
[3] B.-S. Du: “A Simple Proof of Sharkovsky’s Theorem”, Amer. Math. Monthly, Vol. 111(7), (2004), pp. 595–599. | Zbl 1187.37054
[4] S. Elayadi: “On a Converse of Sharkovsky’s Theorem”, Amer. Math. Monthly, Vol. 103, (1996), pp. 386–392. | Zbl 0893.58024
[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. | Zbl 0930.54034
[6] T.-Y. Li and J.A. Yorke: “Period Three Impies Chaos”, Amer. Math. Monthly, Vol. 82(10), (1975), pp. 985–992. | Zbl 0351.92021
[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 | Zbl 0877.26005
[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. | Zbl 0455.58022