Expansive homeomorphisms of solenoidal groups
This paper is concerned with a conjugacy problem for homeomorphisms and endomorphisms on compact connected finite-dimensional abelian groups, which are known as solenoidal groups. Let \(X\) be a solenoidal group and \(f\) a local homeomorphism on \(X\) fixing the identity of \(X\). It was known that there is a totally disconnected subgroup \(F\) such that \(X/F\) is an \(n\)-torus, and such that for each \(\lambda>0\), there exist a continuous map \(\varphi_ f: F\times X\to X\) with \(\text{diam}(\varphi_ f(F\times X))<\lambda\) and a group endomorphism \(\sigma_ f:X\to X\), such that \(f(x+y)=f(x)+\varphi_ f(y,x)+\sigma_ f(y)\) for any \(x\in X\), \(y\in Y\). Furthermore, there exist a continuous homomorphism \(\psi: \mathbb{R}^ n\to X\), a biuniformly continuous bijection \(\bar f:\mathbb{R}^ n\to\mathbb{R}^ n\), a subgroup \(C\) of finite index in \(\mathbb{Z}^ n\), a continuous map \(\bar\varphi_ f: C\times\mathbb{R}^ n\to B(0,\lambda)\) and a linear map \(\bar\sigma_ f:\mathbb{R}^ n\to\mathbb{R}^ n\) such that \(\psi\circ f=f\circ\psi\), \(\psi\circ\bar\sigma_ f=\sigma_ f\circ\psi\), and \(\bar f(v+l)=\bar f(v)+\bar\varphi_ f(l,v)+\bar\sigma_ f(l)\) for all \(v\in\mathbb{R}^ n\), \(l\in\mathbb{C}\). The authors show that if \(f\) is an expansive homeomorphism and has the pseudo-orbit tracing property, \(\bar\sigma_ f\) is hyperbolic and the expansion constant of \(\bar f\) is arbitrary, then \(f\) is topologically conjugate to \(\sigma_ f\), which in the present case is a solenoidal group automorphism. In the case \(X\) is one-dimensional, if \(f\) has a fixed point, then the assumptions on \(\bar\sigma_ f\) and \(\bar f\) can be dropped.
scientific article; zbMATH DE number 3968