Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

• Autorzy: Hisao Kato
• Języki publikacji: EN

Abstrakty

EN

A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that $d(f^n(x), f^n(y)) > c$ (resp. $diam f^n(A) > c$). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate subcontinuum of X satisfying: (i) for each x ∈ Z, $V^σ(x; Z)$ is dense in Z, and (ii) there exists τ > 0 such that for each x ∈ Z and each neighborhood U of x in X, there is y ∈ U ∩ Z such that $lim inf_{n → ∞} d(f^n(x), f^n(y))$ ≥ τ if σ = s, and $lim inf_{n → ∞} d(f^{-n}(x), f^{-n}(y))$ ≥ τ if σ = u; in particular, $W^σ(x) ≠ W^σ(y)$. Here
  $V^s(x; Z) = {z ∈ Z|$ there is a subcontinuum A of Z such that
      x, z ∈ A and $lim_{n → ∞} diam f^n(A) = 0}$,
$V^u(x; Z) = {z ∈ Z| there is a subcontinuum A of Z such that
      x, z ∈ A and $lim_{n → ∞} diam f^{-n}(A) = 0}$,
   $W^s(x) = {x' ∈ X|$ $lim_{n → ∞} d(f^n(x), f^n(x')) = 0}$, and
   $W^u(x) = {x' ∈ X|$ $lim_{n → ∞} d(f^{-n}(x), f^{-n}(x'))=0}$.
As a corollary, if f is a continuum-wise expansive homeomorphism of a compactum X with dim X > 0 and Z is a σ-chaotic continuum of f, then for almost all Cantor sets C ⊂ Z, f or $f^{-1}$ is chaotic on C in the sense of Li and Yorke according as σ = s or u). Also, we prove that if f is a continuum-wise expansive homeomorphism of a compactum X with dim X > 0 and there is a finite family $\mathbb{F}$ of graphs such that X is $\mathbb{F}$-like, then each chaotic continuum of f is indecomposable. Note that every expansive homeomorphism is continuum-wise expansive.

Słowa kluczowe

EN

expansive homeomorphism; continuum-wise expansive homeomorphism; stable and unstable sets; scrambled set; chaotic in the sense of Li and Yorke; independent; indecomposable continuum

Kategorie tematyczne

• 54H20: Topological dynamics
• 54F50: Spaces of dimension ≤ 1 ; curves, dendrites
• 54B20: Hyperspaces
• 54E40: Special maps on metric spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1994
• Tom: 145
• Numer: 3
• Strony: 261-279

Daty

wydano

1994

otrzymano

1993-05-17

poprawiono

1994-01-15

Twórcy

autor

Hisao Kato

• Faculty of Integrated, Arts and Sciences, Hiroshima University, 1-7-1 Kagamiyama, Higashi-Hiroshima, 724 Japan
• Institute of Mathematics, University of Tsukuba, Ibaraki 305, 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] S. B. Curry, One-dimensional nonseparating plane continua with disjoint ε-dense subcontinua, Topology Appl. 39 (1991), 145-151.
• [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] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton, N.J., 1948.
• [9] J. F. Jacobson and W. R. Utz, The nonexistence of expansive homeomorphisms of a closed 2-cell, Pacific J. Math. 10 (1960), 1319-1321.
• [10] H. Kato, The nonexistence of expansive homeomorphisms of Peano continua in the plane, Topology Appl. 34 (1990), 161-165.
• [11] H. Kato, On expansiveness of shift homeomorphisms in inverse limits of graphs, Fund. Math. 137 (1991), 201-210.
• [12] H. Kato, The nonexistence of expansive homeomorphisms of dendroids, ibid. 136 (1990), 37-43.
• [13] H. Kato, Embeddability into the plane and movability on inverse limits of graphs whose shift maps are expansive, Topology Appl. 43 (1992), 141-156.
• [14] H. Kato, Expansive homeomorphisms in continuum theory, ibid. 45 (1992), 223-243.
• [15] H. Kato, Expansive homeomorphisms and indecomposability, Fund. Math. 139 (1991), 49-57.
• [16] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598.
• [17] H. Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topology Appl. 53 (1993), 239-258.
• [18] H. Kato, Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets, Fund. Math. 143 (1993), 153-165.
• [19] H. Kato and K. Kawamura, A class of continua which admit no expansive homeomorphisms, Rocky Mountain J. Math. 22 (1992), 645-651.
• [20] K. Kuratowski, Topology, Vol. II, Academic Press, New York, 1968.
• [21] K. Kuratowski, Applications of Baire-category method to the problem of independent sets, Fund. Math. 81 (1974), 65-72.
• [22] T. Y Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992.
• [23] R. Ma né, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319.
• [24] S. B. Nadler, Jr., Hyperspaces of Sets, Pure and Appl. Math. 49, Dekker, New York, 1978.
• [25] R. V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Math. USSR-Sb. 23 (1974), 233-253.
• [26] W. Reddy, The existence of expansive homeomorphisms of manifolds, Duke Math. J. 32 (1965), 627-632.
• [27] W. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774.
• [28] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math. 79, Springer, 1982.
• [29] R. F. Williams, A note on unstable homeomorphisms, Proc. Amer. Math. Soc. 6 (1955), 308-309.