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1
Content available remote

The nonexistence of expansive homeomorphisms of chainable continua

100%
Kato H.
Fundamenta Mathematicae
|
1996
|
tom 149
|
nr 2
119-126
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 $d(f^n(x),f^n(y)) > c$. In this paper, we prove that if a homeomorphism f:X → X of a continuum X can be lifted to an onto map h:P → P of the pseudo-arc P, then f is not expansive. As a corollary, we prove that there are no expansive homeomorphisms on chainable continua. This is an affirmative answer to one of Williams' conjectures.
2
Content available remote

On indecomposability and composants of chaotic continua

100%
Kato H.
Fundamenta Mathematicae
|
1996
|
tom 150
|
nr 3
245-253
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 $V^σ(z;Z)$.
3
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A note on f.p.p. and $f^*.p.p.$

100%
Kato H.
Colloquium Mathematicum
|
1993
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tom 66
|
nr 1
147-150
EN
In [3], Kinoshita defined the notion of $f^*.p.p.$ and he proved that each compact AR has $f^*.p.p.$ In [4], Yonezawa gave some examples of not locally connected continua with f.p.p., but without $f^*.p.p.$ In general, for each n=1,2,..., there is an n-dimensional continuum $X_n$ with f.p.p., but without $f^*.p.p.$ such that $X_n$ is locally (n-2)-connected (see [4, Addendum]). In this note, we show that for each n-dimensional continuum X which is locally (n-1)-connected, X has f.p.p. if and only if X has $f^*.p.p.$
4
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Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets

100%
Kato H.
Fundamenta Mathematicae
|
1993
|
tom 143
|
nr 2
153-165
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 $dim W^s(W^x)=0$.
5
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Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

100%
Kato H.
Fundamenta Mathematicae
|
1994
|
tom 145
|
nr 3
261-279
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.
6
Content available remote

Finite-dimensional maps and dendrites with dense sets of end points

64%
Kato H., Matsuhashi E.
Colloquium Mathematicum
|
2006
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tom 106
|
nr 1
83-91
EN
The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space $C(X,I^{p+2k+1-i})$ such that the diagonal product $f×g: X → Y×I^{p+2k+1-i}$ is an (i+1)-to-1 map is a dense $G_{δ}$-subset of $C(X,I^{p+2k+1-i})$. In this paper, we prove that if f: X → Y is as above and $D_{j}$ (j = 1,..., k) are superdendrites, then the set of maps h in $C(X,∏_{j=1}^{k} D_{j}×I^{p+1-i})$ such that $f×h: X → Y×(∏_{j=1}^{k} D_{j}×I^{p+1-i})$ is (i+1)-to-1 is a dense $G_{δ}$-subset of $C(X,∏_{j=1}^{k} D_{j}×I^{p+1-i})$ for each 0 ≤ i ≤ p.
7
Content available remote

On Surjective Bing Maps

64%
Kato H., Matsuhashi E.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
|
tom 52
|
nr 3
329-333
EN
In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense $G_δ$-subset of the space of all maps. In [6], J. Krasinkiewicz independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n ≥ 1) is a dense $G_δ$-subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate connected polyhedron is a dense $G_δ$-subset of the space of maps. In this note, we investigate the existence of surjective Bing maps from continua to polyhedra.
8
Content available remote

Whitney continua of graphs admit all homotopy types of compact connected ANRs

44%
Kato H.
Fundamenta Mathematicae
|
1988
|
tom 129
|
nr 3
161-166
9
Content available remote

On expansiveness of shift homeomorphisms of inverse limits of graphs

44%
Kato H.
Fundamenta Mathematicae
|
1991
|
tom 137
|
nr 3
201-210
10
Content available remote

The nonexistence of expansive homeomorphisms of dendroids

38%
Kato H.
Fundamenta Mathematicae
|
1990
|
tom 136
|
nr 1
37-43
11
Content available remote

On local 1-connectedness of Whitney continua

32%
Kato H.
Fundamenta Mathematicae
|
1988
|
tom 131
|
nr 3
245-253
12
Content available remote

Refinable maps in the theory of shape

26%
Kato H.
Fundamenta Mathematicae
|
1981
|
tom 113
|
nr 2
119-129
13
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Content available

On admissible Whitney maps

26%
Kato H.
Colloquium Mathematicum
|
1988
|
tom 56
|
nr 2
299-309
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