The OE-property of group automorphisms
Let \(\phi\) be a self-homeomorphism of a compact metric space (X,d). Say that (X,\(\phi)\) has P.O.T.P. if, for every \(\epsilon >0\), there exists \(\delta >0\) such that, for any \(\delta\)-orbit [see \textit{F. Takens}, Dynamical systems, Proc. Symp. Univ. Warwick, Coventry, 1973/74, Lect. Notes Math. 468, 293-304 (1975; Zbl 0321.54022)], there is a point \(x\in X\) with \(d(\phi^ i(x),x_ i)<\epsilon\) for all i; and say that (X,\(\phi)\) has the weaker property OE if \(O_{\phi}=E_{\phi}\) [see \textit{Takens}, op. cit.] \((O_{\phi}\) denotes the closure in the Hausdorff metric of the set of closures in X of orbits of \(\phi)\). The authors prove, by an elaborate argument depending repeatedly on the structure theory of topological groups, that the properties P.O.T.P. and OE are equivalent when X is a compact metric group and \(\phi\) a homeomorphic automorphism. If X is in addition connected, and (X,\(\phi)\) has OE, then it is topologically transitive. The authors note further that, for automorphisms of finite-dimensional tori, OE is equivalent both to topological stability and to hyperbolicity [see the first author, Nagoya Math. J. 90, 119-135 (1983; Zbl 0507.58030); correction, ibid. 95, 103 (1984)].
self-homeomorphism of a compact metric space
automorphisms of finite-dimensional tori
scientific article; zbMATH DE number 3911277