On exact toral endomorphisms (Q690369)

A continuous endomorphism \(\varphi\) of a compact abelian group \(G\) with normalized Haar measure \(\mu\) is called exact if \(\mu(\bigcap_{n\geq 1} \varphi^{-n}(S))\in \{0,1\}\) for all measurable sets \(S\subset G\). The first part of this paper collects some facts about exact endomorphisms on general compact abelian groups. The second part deals with the \(k\)- dimensional torus \(T^ k=\mathbb{R}^ k/ \mathbb{Z}^ k\). In this case, each endomorphism \(\varphi\) of \(T^ k\) admits a unique regular matrix \(A\) over \(\mathbb{Z}\) of degree \(k\) with \(\varphi(x)={Ax}\pmod{\mathbb{Z}^ k}\). The main result of this paper is a necessary and sufficient condition for \(\varphi\) being exact in terms of the characteristic polynomial of \(A\).