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A chainable continuum, X, and homeomorphisms, p,q: X → X, are constructed with the following properties:
(1) p ∘ q = q ∘ p,
(2) p, q have simple dynamics,
(3) p ∘ q is a positively continuum-wise fully expansive homeomorphism that has positive entropy and is chaotic in the sense of Devaney and in the sense of Li and Yorke.
A homeomorphism h:X → X of a compactum X is expansive provided that for some fixed c > 0 and any distinct x, y ∈ X there exists an integer n, dependent only on x and y, such that d(hⁿ(x),hⁿ(y)) > c. It is shown that if X is a circle-like continuum that admits an expansive homeomorphism, then X is homeomorphic to a solenoid.
A homeomorphism h: X → X of a compactum X is expansive provided that for some fixed c > 0 and every x, y ∈ X (x ≠ y) there exists an integer n, dependent only on x and y, such that d(hⁿ(x),hⁿ(y)) > c. It is shown that if X is a solenoid that admits an expansive homeomorphism, then X is homeomorphic to a regular solenoid. It can then be concluded that a circle-like continuum admits an expansive homeomorphism if and only if it is homeomorphic to a regular solenoid.
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