Growth degree classification for finitely generated semigroups of integer matrices (Q284638)

Let \(\mathcal{A} = \{ {A_1},\dots{A_m}\} \) be a finite set of matrices, \({m_n}(\mathcal{A})\) be the maximum norm of a product of \(n\) elements of \(\mathcal{A}\); the set \(\mathcal{A}\) is called non-degenerate if \({m_n}(\mathcal{A}) \nrightarrow 0\). It is proved that if \(\mathcal{A}\) is a finite non-degenerate set of \(d \times d\) integer matrices then \(\mathop {k = \lim }\limits_{n \to \infty } \frac{{\log {m_n}(\mathcal{A})}}{{\log n}} \in {\mathbb{Z}_{ \geqslant 0}} \cup \{ \infty \} \) and \(k \in {\mathbb{Z}_{ \geqslant 0}}\) iff the joint spectral radius \(\rho \left( \mathcal{A} \right) = 1\). In this case, there are positive constants \({C_1},{C_2}\) such that \({C_1}{n^k} \leqslant {m_n}(\mathcal{A}) \leqslant {C_2}{n^k}\) for all \(n \geqslant 1\) (the joint spectral radius \(\rho (\mathcal{A}) = \mathop {\lim }\limits_{n \to \infty } |{m_n}(\mathcal{A}){|^{1/n}}\)).