# A topological invariant for pairs of maps

Marcelo Polezzi; Claudemir Aniz

Open Mathematics (2006)

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

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topMarcelo 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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- [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).
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- [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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