Flow equivalence of translations on compact metric Abelian groups (Q1262399)

Let \(\Gamma\) {\^{\ }} be the character group of a countable discrete subgroup \(\Gamma\) of the unit disc group, for \(\chi\in \Gamma {\hat{\;}}\) and \(z\in \Gamma\), let R(\(\Gamma)\) be a homeomorphism of \(\Gamma\) {\^{\ }} defined by \(R(\Gamma)\chi =\chi \chi_{\Gamma}\), where \(\chi_{\Gamma}\in \Gamma {\hat{\;}}\) is determined by \(<z,\chi_{\Gamma}>=z\). Thus R(\(\Gamma)\) can be interpreted as a translation of \(\Gamma\) {\^{\ }}. In this paper, the authors prove that \(R(\Gamma_ 1)\) and \(R(\Gamma_ 2)\) are mutually flow equivalent if and only if \(K_ 1=cK_ 2\), where c is a positive constant and \(K_ j\) are subgroups of the real group such that \(K_ i=\{x\in R|\) \(\exp (2\pi ix)\in \Gamma_ j\}\), \(j=1,2.\) In the proof of the above result, the authors observe that \(R(\Gamma_ 1)\) and \(R(\Gamma_ 2)\) are mutually ergodic and the flow \(((R(\Gamma_ 1),1)_ t)_{t\in R}\) is topologically conjugate with \(((R(\Gamma_ 2),\frac{1}{c})_ t)_{t\in R}\). The proof depends on the use of Schwartzman's winding numbers. This result can be applied to flow equivalence of n-dimensional irrational rotations, adding machine transformations and solenoidal transformations respectively.