Minimal sets and Chaos in the Sense of Devaney on Continuum-Wise Expansive Homeomorphisms
Hisao Kato · Continua. · 2020
In [ 15 ], we proved that if a homeomorphism f : X → X of a compactum X is continuum-wise expansive and dim X > 0, then there is a chaotic continuum Z of f and either f or f –1 is chaotic on almost all Cantor sets of Z in the sense of Li-Yorke. In [ 16 ], for a map f : X → X , we defined the family D ( f ) = ( ℳ + ( f ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072379/ca24cf38-3870-4983-95dc-9b04e0841f90/content/eq3044.tif"/> consisting of all minimal elements of not zero-dimensional, f -invariant closed subsets of X , and we showed that if f : X → X is a continuum-wise expansive homeomorphism of a compactum X with dim X > 0, then D ( f ) ≠ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072379/ca24cf38-3870-4983-95dc-9b04e0841f90/content/eq3045.tif"/> and if Y ∈ D ( f ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072379/ca24cf38-3870-4983-95dc-9b04e0841f90/content/eq3046.tif"/> , both f | Y : Y → Y and f –1 | Y : Y → Y are sensitive and topologically transitive. In this paper, we study the minimal sets of continuum-wise expansive homeomorphisms. In particular, we show that if a homeomorphism f : X → X of a compactum X is continuum-wise expansive and dim X > 0, then both f | Y : Y → Y and f –1 | Y → Y are weakly chaotic in the sense of Devaney for each Y ∈ D ( f ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072379/ca24cf38-3870-4983-95dc-9b04e0841f90/content/eq3047.tif"/> We know that several chaotic properties are concentrated on such sets Y ∈ D ( f ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072379/ca24cf38-3870-4983-95dc-9b04e0841f90/content/eq3048.tif"/> .