A topological invariant for pairs of maps

Marcelo Polezzi; Claudemir Aniz

Open Mathematics (2006)

  • Volume: 4, Issue: 2, page 294-303
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Marcelo Polezzi, and Claudemir Aniz. "A topological invariant for pairs of maps." Open Mathematics 4.2 (2006): 294-303. <http://eudml.org/doc/269052>.

@article{MarceloPolezzi2006,
abstract = {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.},
author = {Marcelo Polezzi, Claudemir Aniz},
journal = {Open Mathematics},
keywords = {26A18; 37E05},
language = {eng},
number = {2},
pages = {294-303},
title = {A topological invariant for pairs of maps},
url = {http://eudml.org/doc/269052},
volume = {4},
year = {2006},
}

TY - JOUR
AU - Marcelo Polezzi
AU - Claudemir Aniz
TI - A topological invariant for pairs of maps
JO - Open Mathematics
PY - 2006
VL - 4
IS - 2
SP - 294
EP - 303
AB - 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.
LA - eng
KW - 26A18; 37E05
UR - http://eudml.org/doc/269052
ER -

References

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  1. [1] R. Barton and K. Burns: “A Simple Special Case of Sharkovskii’s Theorem”, Amer. Math. Monthly, Vol. 107(10), (2000), pp. 932–933. Zbl0979.37016
  2. [2] N. Bhatia: “New Proof and Extension of Sarkovskii’s Theorem”, Far East J. Math. Sci., Special Volume, Part I, (1996), pp. 53–68. Zbl0928.54038
  3. [3] B.-S. Du: “A Simple Proof of Sharkovsky’s Theorem”, Amer. Math. Monthly, Vol. 111(7), (2004), pp. 595–599. Zbl1187.37054
  4. [4] S. Elayadi: “On a Converse of Sharkovsky’s Theorem”, Amer. Math. Monthly, Vol. 103, (1996), pp. 386–392. Zbl0893.58024
  5. [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. Zbl0930.54034
  6. [6] T.-Y. Li and J.A. Yorke: “Period Three Impies Chaos”, Amer. Math. Monthly, Vol. 82(10), (1975), pp. 985–992. Zbl0351.92021
  7. [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 Zbl0877.26005
  8. [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. [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. [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. [11] P.D. Straffin, Jr.: “Periodic Points of Continuous Functions”, Math. Mag., Vol. 51(2), (1978), pp. 99–105. Zbl0455.58022

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