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## Fundamenta Mathematicae

1993 | 143 | 2 | 153-165
Tytuł artykułu

### Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets

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Języki publikacji
EN
Abstrakty
EN
We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dim X > 0, then the decompositions ${W^S(x)|x ∈ X}$ and ${W^(u)(x)| x ∈ X}$ of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, $W^σ(x)$ contains a nondegenerate subcontinuum $A_x$ containing x with $diam A_x ≥ ϱ$, and if x,y ∈ C and x ≠ y, then $W^σ(x) ≠ W^σ(y)$. For a continuum-wise expansive homeomorphism, a similar result is obtained. Also, we prove that if f : G → G is a map of a graph G and the shift map ˜f: (G,f) → (G,f) of f is expansive, then for each ˜x ∈ (G,f), $W^u(˜x)$ is equal to the arc component of (G,f) containing ˜x, and $dim W^s(W^x)=0$.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
153-165
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-11-19
poprawiono
1993-03-15
poprawiono
1993-04-06
Twórcy
autor
• Faculty of Integrated Arts and Sciences. Hiroshima University, 1-7-1 Kagamiyama, Higashi-Hiroshima, 724 Japan
Bibliografia
• [1] N. Aoki, Topological dynamics, in: Topics in General Topology, K. Morita and J. Nagata (eds.), Elsevier, 1989, 625-740.
• [2] B. F. Bryant, Unstable self-homeomorphisms of a compact space, thesis, Vanderbilt University, 1954.
• [3] M. Denker and M. Urbański, Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3 (1991), 561-579.
• [4] R. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, 1989.
• [5] W. Gottschalk, Minimal sets; an introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336-351.
• [6] W. Gottschalk and G. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ. 34, Amer. Math. Soc., 1955.
• [7] K. Hiraide, Expansive homeomorphisms on compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), 117-162.
• [8] J. F. Jacobson and W. R. Utz, The nonexistence of expansive homeomorphisms of a closed 2-cell, Pacific J. Math. 10 (1960), 1319-1321.
• [9] H. Kato, The nonexistence of expansive homeomorphisms of Peano continua in the plane, Topology Appl. 34 (1990), 161-165.
• [10] H. Kato, On expansiveness of shift homeomorphisms of inverse limits of graphs, Fund. Math. 137 (1991), 201-210.
• [11] H. Kato, The nonexistence of expansive homeomorphisms of dendroids, ibid. 136 (1990), 37-43.
• [12] H. Kato, Embeddability into the plane and movability on inverse limits of graphs whose shift maps are expansive, Topology Appl. 43 (1992), 141-156.
• [13] H. Kato, Expansive homeomorphisms in continuum theory, ibid. 45 (1992), 223-243.
• [14] H. Kato, Expansive homeomorphisms and indecomposability, Fund. Math. 139 (1991), 49-57.
• [15] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598.
• [16] H. Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topology Appl., to appear.
• [17] H. Kato and K. Kawamura, A class of continua which admit no expansive homeomorphisms, Rocky Mountain J. Math. 22 (1992), 645-651.
• [18] K. Kuratowski, Topology, Vol. II, Academic Press, New York, 1968.
• [19] K. Kuratowski, Applications of Baire-category method to the problem of independent sets, Fund. Math. 81 (1974), 65-72.
• [20] R. Ma né, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319.
• [21] S. B. Nadler, Jr., Hyperspaces of Sets, Pure and Appl. Math. 49, Dekker, New York, 1978.
• [22] R. V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Math. USSR-Sb. 23 (1974), 233-253.
• [23] R. V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Surveys 39 (1984), 85-131.
• [24] W. Reddy, The existence of expansive homeomorphisms of manifolds, Duke Math. J. 32 (1965), 627-632.
• [25] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer, 1982.
• [26] R. F. Williams, A note on unstable homeomorphisms, Proc. Amer. Math. Soc. 6 (1955), 308-309.
• [27] R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473-487.
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Bibliografia
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