The dynamics of homeomorphisms of hereditarily decomposable chainable continua (Q1896638)

A continuum \(M\) is said to be hereditarily decomposable if each subcontinuum can be decomposed as the union of two proper subcontinua and is said to be chainable (respectively circle-like) if for each \(\varepsilon > 0\), there exists a continuous map \(f_\varepsilon : M \to [0,1]\) (respectively \(S^1\)) such that \(\text{diam} (f_\varepsilon^{-1} (t)) < \varepsilon\) for each \(t\). The author shows that if \(M\) is a hereditarily decomposable chainable continuum and \(F : M \to M\) is a homeomorphism, then any periodic orbit of \(F\) has period a power of 2 and for each \(x \in M\) which is recurrent, the \(\omega\)-limit set of \(x\) is either a periodic orbit of \(F\) or is a subset of \(M\) on which \(F\) acts as a dyadic adding machine. The author deduces that homeomorphisms of chainable and circle-like Suslinean continua have zero topological entropy in support of a conjecture of M. Barge that all homeomorphisms of all Suslinean continua have zero topological entropy.