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

Fundamenta Mathematicae (1994)

- Volume: 145, Issue: 3, page 261-279
- ISSN: 0016-2736

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topKato, Hisao. "Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke." Fundamenta Mathematicae 145.3 (1994): 261-279. <http://eudml.org/doc/212046>.

@article{Kato1994,

abstract = {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 $limn → ∞ diam fn(A) = 0$,
$$V^u(x; Z) = \{z ∈ Z| there is a subcontinuum A of Z such that
\}~~~~~ x, z ∈ A and $limn → ∞ diam f-n(A) = 0$,
$ $W^s(x) = \{x^\{\prime \} ∈ X| lim_\{n → ∞\} d(f^n(x), f^n(x^\{\prime \})) = 0\}$, and
$W^u(x) = \{x^\{\prime \} ∈ X| lim_\{n → ∞\} d(f^\{-n\}(x), f^\{-n\}(x^\{\prime \}))=0\}$.
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.
},

author = {Kato, Hisao},

journal = {Fundamenta Mathematicae},

keywords = {expansive homeomorphism; continuum-wise expansive homeomorphism; stable and unstable sets; scrambled set; chaotic in the sense of Li and Yorke; independent; indecomposable continuum; chaos},

language = {eng},

number = {3},

pages = {261-279},

title = {Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke},

url = {http://eudml.org/doc/212046},

volume = {145},

year = {1994},

}

TY - JOUR

AU - Kato, Hisao

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

JO - Fundamenta Mathematicae

PY - 1994

VL - 145

IS - 3

SP - 261

EP - 279

AB - 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 $limn → ∞ diam fn(A) = 0$,
$$V^u(x; Z) = {z ∈ Z| there is a subcontinuum A of Z such that
}~~~~~ x, z ∈ A and $limn → ∞ diam f-n(A) = 0$,
$ $W^s(x) = {x^{\prime } ∈ X| lim_{n → ∞} d(f^n(x), f^n(x^{\prime })) = 0}$, and
$W^u(x) = {x^{\prime } ∈ X| lim_{n → ∞} d(f^{-n}(x), f^{-n}(x^{\prime }))=0}$.
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.

LA - eng

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

UR - http://eudml.org/doc/212046

ER -

## References

top- [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. Zbl0718.54042
- [4] R. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, 1989. Zbl0695.58002
- [5] W. Gottschalk, Minimal sets: an introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336-351. Zbl0085.17401
- [6] W. Gottschalk and G. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ. 34, Amer. Math. Soc., 1955. Zbl0067.15204
- [7] K. Hiraide, Expansive homeomorphisms on compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), 117-162. Zbl0713.58042
- [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. Zbl0144.22302
- [10] H. Kato, The nonexistence of expansive homeomorphisms of Peano continua in the plane, Topology Appl. 34 (1990), 161-165. Zbl0713.54035
- [11] H. Kato, On expansiveness of shift homeomorphisms in inverse limits of graphs, Fund. Math. 137 (1991), 201-210. Zbl0738.54017
- [12] H. Kato, The nonexistence of expansive homeomorphisms of dendroids, ibid. 136 (1990), 37-43. Zbl0707.54028
- [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. Zbl0767.54008
- [14] H. Kato, Expansive homeomorphisms in continuum theory, ibid. 45 (1992), 223-243. Zbl0790.54048
- [15] H. Kato, Expansive homeomorphisms and indecomposability, Fund. Math. 139 (1991), 49-57. Zbl0823.54028
- [16] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598. Zbl0797.54047
- [17] H. Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topology Appl. 53 (1993), 239-258. Zbl0797.54048
- [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. Zbl0790.54053
- [19] H. Kato and K. Kawamura, A class of continua which admit no expansive homeomorphisms, Rocky Mountain J. Math. 22 (1992), 645-651. Zbl0823.54029
- [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. Zbl0311.54036
- [22] T. Y Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. Zbl0351.92021
- [23] R. Ma né, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319. Zbl0362.54036
- [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. Zbl0324.58013
- [26] W. Reddy, The existence of expansive homeomorphisms of manifolds, Duke Math. J. 32 (1965), 627-632. Zbl0132.18904
- [27] W. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. Zbl0040.09903
- [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. Zbl0067.15402

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