On a certain map of a triangle

Grzegorz Świrszcz

Fundamenta Mathematicae (1998)

  • Volume: 155, Issue: 1, page 45-57
  • ISSN: 0016-2736

Abstract

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The paper answers some questions asked by Sharkovski concerning the map F:(u,v) ↦ (u(4-u-v),uv) of the triangle Δ = u,v ≥ 0: u+v ≤ 4. We construct an absolutely continuous σ-finite invariant measure for F. We also prove the following strange phenomenon. The preimages of side I = Δ ∩ v=0 form a dense subset ∪ F - n ( I ) of Δ and there is another dense set Λ consisting of points whose orbits approach the interval I but are not attracted by I.

How to cite

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Świrszcz, Grzegorz. "On a certain map of a triangle." Fundamenta Mathematicae 155.1 (1998): 45-57. <http://eudml.org/doc/212242>.

@article{Świrszcz1998,
abstract = {The paper answers some questions asked by Sharkovski concerning the map F:(u,v) ↦ (u(4-u-v),uv) of the triangle Δ = u,v ≥ 0: u+v ≤ 4. We construct an absolutely continuous σ-finite invariant measure for F. We also prove the following strange phenomenon. The preimages of side I = Δ ∩ v=0 form a dense subset $∪F^\{-n\}(I)$ of Δ and there is another dense set Λ consisting of points whose orbits approach the interval I but are not attracted by I.},
author = {Świrszcz, Grzegorz},
journal = {Fundamenta Mathematicae},
keywords = {non-hyperbolic dynamical systems; invariant measure; attractors},
language = {eng},
number = {1},
pages = {45-57},
title = {On a certain map of a triangle},
url = {http://eudml.org/doc/212242},
volume = {155},
year = {1998},
}

TY - JOUR
AU - Świrszcz, Grzegorz
TI - On a certain map of a triangle
JO - Fundamenta Mathematicae
PY - 1998
VL - 155
IS - 1
SP - 45
EP - 57
AB - The paper answers some questions asked by Sharkovski concerning the map F:(u,v) ↦ (u(4-u-v),uv) of the triangle Δ = u,v ≥ 0: u+v ≤ 4. We construct an absolutely continuous σ-finite invariant measure for F. We also prove the following strange phenomenon. The preimages of side I = Δ ∩ v=0 form a dense subset $∪F^{-n}(I)$ of Δ and there is another dense set Λ consisting of points whose orbits approach the interval I but are not attracted by I.
LA - eng
KW - non-hyperbolic dynamical systems; invariant measure; attractors
UR - http://eudml.org/doc/212242
ER -

References

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  1. [1] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Basel, 1980. Zbl0458.58002
  2. [2] A. Douady et J. Hubbard, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris 294 (1982), 123-126. 
  3. [3] R. Lozi, Un attracteur étrange du type attracteur de Hénon, J. Phys. 39 (1978), suppl. au no. 8, 9-10. 

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