1993
DOI: 10.4153/cjm-1993-030-4
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Continuum-Wise Expansive Homeomorphisms

Abstract: The notion of expansive homeomorphism is important in topological dynamics and continuum theory. In this paper, a new kind of homeomorphism will be introduced and studied, namely the continuum-wise expansive homeomorphism. The class of continuum-wise expansive homeomorphisms is much larger than the one of expansive homeomorphisms. In fact, the class of continuum-wise expansive homeomorphisms contains many important homeomorphisms which often appear in "chaotic" topological dynamics and continuum theory, but wh… Show more

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Cited by 125 publications
(116 citation statements)
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“…The proofs of the following theorems are exactly the same as the proofs for Corollary 2.4 and Proposition 2.5 in [2]. However the following theorems are stated for compacta, whereas Corollary 2.4 and Proposition 2.5 are stated for continua.…”
Section: Entropy and Expansive Homeomorphismsmentioning
confidence: 56%
See 1 more Smart Citation
“…The proofs of the following theorems are exactly the same as the proofs for Corollary 2.4 and Proposition 2.5 in [2]. However the following theorems are stated for compacta, whereas Corollary 2.4 and Proposition 2.5 are stated for continua.…”
Section: Entropy and Expansive Homeomorphismsmentioning
confidence: 56%
“…However, the infinite folding also raises the entropy of the homeomorphism, and it is known that expansive homeomorphisms must have finite entropy. In this paper it will be shown that continuum-wise expansive homeomorphisms on finite-dimensional hereditarily indecomposable compacta must have infinite entropy [2]. It will then be concluded that hereditarily indecomposable compacta cannot admit expansive homeomorphisms.…”
Section: Introductionmentioning
confidence: 99%
“…These ideas will be extended for cw-expansive homeomorphisms on a compact metric space. Recall from [2,3] that a homeomorphism f : X → X is cw-expansive if there is δ > 0 such that if A ⊂ X is connected with more than one point then there is n ∈ Z such that diam(f n (A)) > δ. This paper is organized as follows.…”
Section: Introductionmentioning
confidence: 99%
“…Theorem 13 ( [8]). If h : X → X is an expansive homeomorphism, then there exists either a stable or an unstable subcontinuum.…”
Section: Suppose On the Contrary Thatmentioning
confidence: 99%