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

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 ${\displaystyle d(f^n\left(x\right),f^n\left(y\right))>c}$ (resp. ${\displaystyle diam{f}^n\left(A\right)>c}$). 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 of X satisfying: (i) for each x ∈ Z, ${\displaystyle V^{\sigma }(x;Z)}$ 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 ${\displaystyle lim\in f_{n\to \infty }d(f^n\left(x\right),f^n\left(y\right))}$ ≥ τ if σ = s, and ${\displaystyle lim\in f_{n\to \infty }d(f^{-n}\left(x\right),f^{-n}\left(y\right))}$ ≥ τ if σ = u; in particular, ${\displaystyle W^{\sigma }\left(x\right)\ne W^{\sigma }\left(y\right)}$. Here   ${\displaystyle V^s(x;Z)=\{z\in Z\mid }$ there is a subcontinuum A of Z such that       x, z ∈ A and ${\displaystyle \underset{n\to \infty }{lim}diam{f}^n\left(A\right)=0\}}$, ${\displaystyle V^u(x;Z)=\{z\in Z\mid thereisa\subset cont\in \cup mAofZsucht\widehat{ } x,z\in A\text{and}}$lim_{n → ∞} diam f^{-n}(A) = 0}${\displaystyle , }$W^s(x) = {x' ∈ X|${\displaystyle }$lim_{n → ∞} d(f^n(x), f^n(x')) = 0}${\displaystyle ,\text{and} }$W^u(x) = {x' ∈ X|${\displaystyle }$lim_{n → ∞} d(f^{-n}(x), f^{-n}(x'))=0}${\displaystyle .Asac\text{or}ollary,iffisacont\in \cup m-wise\exp ansivehomeom\text{or}\varphi smofaco\mp actumXwithdimX>0\text{and}Zisa\sigma -chaoticcont\in \cup moff,thenf\text{or}almostallCan\to rsetsC\subset Z,f\text{or}}$f^{-1}${\displaystyle ischaoticonC\in thesenseofLi\text{and}Y\text{or}kea\text{or}d\in gas\sigma =s\text{or}u).Also,weprovet\widehat{if}fisacont\in \cup m-wise\exp ansivehomeom\text{or}\varphi smofaco\mp actumXwithdimX>0\text{and}thereisaf\in itefamily}$\mathbb{F}${\displaystyle ofgraphssucht\hat{X}is}$\mathbb{F}!$!-like, then each chaotic continuum of f is indecomposable. Note that every expansive homeomorphism is continuum-wise expansive.