On asymptotic properties of matrix semigroups with an invariant cone (Q442681)

Two joint spectral characteristics are studied for finitely generated matrix semigroups, when the matrices share one or two invariant cones. For a bounded set \(\Sigma\in{\mathbb R}^{n\times n}\), these characteristics are the joint spectral radius \(\rho(\Sigma)\) and the joint spectral subradius \(\check{\rho}(\Sigma)\) respectively, NEWLINE\[NEWLINE\rho(\Sigma)=\displaystyle\lim_{t\rightarrow\infty}\sup\{\parallel A_1\cdots A_t\parallel^{1/t}: A_i\in\Sigma\},\;\check{\rho}(\Sigma)=\displaystyle\lim_{t\rightarrow\infty}\inf\{\parallel A_1\cdots A_t\parallel^{1/t}: A_i\in\Sigma\}.NEWLINE\]NEWLINENEWLINENEWLINEIt is well-known that \(\rho(\Sigma)\) is continuous w.r.t. the Hausdorff distance, but \(\check{\rho}(\Sigma)\) is not continuous. In this paper the continuity of the joint spectral subradius is proved in the neighborhood of sets of matrices that leave an embedded pair of cones invariant.NEWLINENEWLINEA convex closed cone \(K'\) is embedded in a cone \(K\subset{\mathbb R}^{n\times n}\) if \((K'\setminus\{0\})\subset\text{int} K\) and in this case one says that \(\{K,K'\}\) is an embedded pair.NEWLINENEWLINEThe main theorem states that if \(\Sigma\) is a compact set in \({\mathbb R}^{n\times n}\) which leaves an embedded pair of cones invariant and \((\Sigma_k)\) is a sequence of sets in \({\mathbb R}^{n\times n}\) that converges to \(\Sigma\) in the Hausdorff metric, then \(\check{\rho}(\Sigma_k)\rightarrow\check{\rho}(\Sigma)\) as \(k\rightarrow\infty\).NEWLINENEWLINEThe author denotes by \(\Sigma^t\) the set of products of length \(t\) of matrices from the bounded set \(\Sigma\) of matrices which leave a cone \(K\) invariant. He proves that if there exists \(A\in \Sigma\) which is \(K\)-primitive (i.e. \(\exists m\in{\mathbb N}\) such that \(A^m(K\setminus\{0\})\subset\text{int} K\)), then both the maximal trace \(\displaystyle\max_{A\in\Sigma^t}\{\text{tr}^{1/t}(A)\}\) and the averaged maximal spectral radius \(\displaystyle\max_{A\in\Sigma^t}\{\rho^{1/t}(A)\}\) converge to the joint spectral radius as \(t\rightarrow\infty\).