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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 $d(f^n\left(x\right),f^n\left(y\right))>c$. 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.

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 $W^S\left(x\right)|x\in X$ and $W^{(}u\left)\left(x\right)\right|x\in X$ 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, $W^{\sigma }\left(x\right)$ contains a nondegenerate...

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 $d(f^n\left(x\right),f^n\left(y\right))>c$ (resp. $diam{f}^n\left(A\right)>c$). 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...

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 $d(f^n\left(x\right),f^n\left(y\right))>c$. 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 $diami{f}^n\left(A\right)>c$. 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...

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 $f^{*}.p.p.$

In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C\left(Q\right)$ and $C\left(\Delta \right)$, where $Q$ and $\Delta$ denote the Hilbert cube ${[0,1]}^{\infty }$ and a Cantor set, respectively.

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 $f\times g:X\to Y\times I^{p+2k+1-i}$ is an (i+1)-to-1 map is a dense $G_{\delta }$-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 $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ such that $f\times h:X\to Y\times \left({\prod }_{j=1}^k{D}_j\times I^{p+1-i}\right)$ is (i+1)-to-1 is a dense $G_{\delta }$-subset of $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ for each 0 ≤ i ≤ p.

In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense $G_{\delta }$-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 $G_{\delta }$-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...