# On indecomposability and composants of chaotic continua

Fundamenta Mathematicae (1996)

- Volume: 150, Issue: 3, page 245-253
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topKato, Hisao. "On indecomposability and composants of chaotic continua." Fundamenta Mathematicae 150.3 (1996): 245-253. <http://eudml.org/doc/212175>.

@article{Kato1996,

abstract = {A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x,y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that $d(f^n(x),f^n(y)) > c$. A homeomorphism f: X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ ℤ such that $diami f^n(A) > c$. Clearly, every expansive homeomorphism is continuum-wise expansive, but the converse assertion is not true. In [6], we defined the notion of chaotic continua of homeomorphisms and proved the existence of chaotic continua of continuum-wise expansive homeomorphisms. Also, we studied indecomposability of chaotic continua. In this paper, we investigate further more properties of indecomposability of chaotic continua and their composants. In particular, we prove that if f:X → X is a continuum-wise expansive homeomorphism of a plane compactum $X ⊂ ℝ^2$ with dim X > 0, then there exists a σ-chaotic continuum Z (σ = s or u) of f such that Z is an indecomposable subcontinuum of X and for each z ∈ Z the composant c(z) of Z containing z coincides with the continuum-wise σ-stable set $V^σ(z;Z)$.},

author = {Kato, Hisao},

journal = {Fundamenta Mathematicae},

keywords = {expansive homeomorphism; continuum-wise expansive homeomorphism; indecomposable; composant; chaotic continuum; plane compactum; stable and unstable sets; stable sets; unstable sets; indecomposability; indecomposable subcontinuum},

language = {eng},

number = {3},

pages = {245-253},

title = {On indecomposability and composants of chaotic continua},

url = {http://eudml.org/doc/212175},

volume = {150},

year = {1996},

}

TY - JOUR

AU - Kato, Hisao

TI - On indecomposability and composants of chaotic continua

JO - Fundamenta Mathematicae

PY - 1996

VL - 150

IS - 3

SP - 245

EP - 253

AB - A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x,y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that $d(f^n(x),f^n(y)) > c$. A homeomorphism f: X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ ℤ such that $diami f^n(A) > c$. Clearly, every expansive homeomorphism is continuum-wise expansive, but the converse assertion is not true. In [6], we defined the notion of chaotic continua of homeomorphisms and proved the existence of chaotic continua of continuum-wise expansive homeomorphisms. Also, we studied indecomposability of chaotic continua. In this paper, we investigate further more properties of indecomposability of chaotic continua and their composants. In particular, we prove that if f:X → X is a continuum-wise expansive homeomorphism of a plane compactum $X ⊂ ℝ^2$ with dim X > 0, then there exists a σ-chaotic continuum Z (σ = s or u) of f such that Z is an indecomposable subcontinuum of X and for each z ∈ Z the composant c(z) of Z containing z coincides with the continuum-wise σ-stable set $V^σ(z;Z)$.

LA - eng

KW - expansive homeomorphism; continuum-wise expansive homeomorphism; indecomposable; composant; chaotic continuum; plane compactum; stable and unstable sets; stable sets; unstable sets; indecomposability; indecomposable subcontinuum

UR - http://eudml.org/doc/212175

ER -

## References

top- [1] N. Aoki, Topological dynamics, in: Topics in General Topology, K. Morita and J. Nagata (eds.), Elsevier, 1989, 625-740.
- [2] J. F. Jacobson and W. R. Utz, The nonexistence of expansive homeomorphisms of a closed 2-cell, Pacific J. Math. 10 (1960), 1319-1321. Zbl0144.22302
- [3] H. Kato, The nonexistence of expansive homeomorphisms of Peano continua in the plane, Topology Appl. 34 (1990), 161-165. Zbl0713.54035
- [4] H. Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topology Appl. 53 (1993), 239-258. Zbl0797.54048
- [5] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598. Zbl0797.54047
- [6] H. Kato, Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke, Fund. Math. 145 (1994), 261-279. Zbl0809.54033
- [7] H. Kato, Chaos of continuum-wise expansive homeomorphisms and dynamical properties of sensitive maps of graphs, Pacific J. Math., to appear. Zbl0865.54039
- [8] K. Kuratowski, Topology, Vol. II, Academic Press, New York, 1968.
- [9] R. Ma né, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319. Zbl0362.54036
- [10] S. B. Nadler, Jr., Hyperspaces of Sets, Pure and Appl. Math. 49, Dekker, New York, 1978.
- [11] T. O'Brien and W. Reddy, Each compact orientable surface of positive genus admits an expansive homeomorphism, Pacific J. Math. 35 (1970), 737-741. Zbl0187.44904
- [12] R. V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Surveys 39 (1984), 85-131. Zbl0584.58038
- [13] W. Reddy, The existence of expansive homeomorphisms of manifolds, Duke Math. J. 32 (1965), 627-632. Zbl0132.18904
- [14] W. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. Zbl0040.09903
- [15] R. F. Williams, A note on unstable homeomorphisms, Proc. Amer. Math. Soc. 6 (1955), 308-309. Zbl0067.15402

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.