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## Fundamenta Mathematicae

1996 | 150 | 3 | 245-253
Tytuł artykułu

### On indecomposability and composants of chaotic continua

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Treść / Zawartość
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EN
Abstrakty
EN
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 $V^σ(z;Z)$.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
245-253
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-06-20
Twórcy
autor
Bibliografia
• [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.
• [3] H. Kato, The nonexistence of expansive homeomorphisms of Peano continua in the plane, Topology Appl. 34 (1990), 161-165.
• [4] H. Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topology Appl. 53 (1993), 239-258.
• [5] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598.
• [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.
• [7] H. Kato, Chaos of continuum-wise expansive homeomorphisms and dynamical properties of sensitive maps of graphs, Pacific J. Math., to appear.
• [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.
• [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.
• [12] R. V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Surveys 39 (1984), 85-131.
• [13] W. Reddy, The existence of expansive homeomorphisms of manifolds, Duke Math. J. 32 (1965), 627-632.
• [14] W. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774.
• [15] R. F. Williams, A note on unstable homeomorphisms, Proc. Amer. Math. Soc. 6 (1955), 308-309.
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Bibliografia
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