Compactness conditions for semigroups of linear maps (Q1337477)

Every compact subsemigroup of the semigroup \(\text{End}(\mathbb{R}^ n)\) is contained in a maximal compact one, and each of these is the full semigroup containing all endomorphisms of \(\mathbb{R}^ n\) mapping the unit ball of some norm into itself. Two of these maximal compact semigroups are isomorphic iff they are conjugate under an automorphism of \(\mathbb{R}^ n\). However, this information says nothing concretely on the relative compactness of the semigroup generated by two endomorphisms of \(\mathbb{R}^ n\) given explicitly in terms of their matrices. It is this situation that is tested and discussed in the paper. The authors consider the subsemigroup \(S\) of \(\text{End}(\mathbb{R}^ n)\) generated by invertible elements \(g = \left( \begin{smallmatrix} a & b\\ c & d\end{smallmatrix} \right)\) and \(h = \left( \begin{smallmatrix} r & 0\\ 0 & s\end{smallmatrix} \right)\) such that \(c \neq 0\), \(0 < r < s < 1\), and they specify explicitly sufficient conditions for \(\overline S\) to be compact with \(\{0 \}\) a minimal ideal. They call such semigroups \(Z\)-semigroups. They employ the homomorphism \(\pi : \text{Gl}(2,\mathbb{R}) \to \text{PSl}(2,\mathbb{R})\) from the group of automorphisms of \(\mathbb{R}^ 2\) to the group of projective automorphisms of the projective real line which they identify with \(\mathbb{R} \cup \infty\). For \(g \in \text{Gl} (2,\mathbb{R})\) there is a well defined function \(z \mapsto \| g(u,v)\| : \mathbb{R} \cup \infty \to \mathbb{R}^ +\) with \(z = u/v\), \(u^ 2 + v^ 2 = 1\). The authors find conditions for \(S\) to be a \(Z\)-semigroup in terms of the action of \(\pi(S)\) on \(\mathbb{R} \cup \infty\). These conditions are explicit but also very technical. The authors also present numerical examples which represent the various case distinctions they have to make and which were found with the aid of a computer search.