On the asymptotic stability of nonnegative matrices in max algebra (Q2568392)

In the max algebra system, for \(n\times n\) real matrices \(A\) and \(B\), the product \(A\otimes B\) has the \((ij)\) entry defined by \(\max_{1\leq k\leq n} a_{ik}b_{kj}\), and for \(x\in\mathbb R^n\), \(A\otimes x\) has the \(i\)th component defined by \(\max_{1\leq j\leq n}a_{ij}x_{j}\). Fix a norm \(| | \cdot| | \) on \(\mathbb R^n\) and define \(\eta (A)=\sup_{| | x| | =1, x\geq 0} | | A\otimes x| | \), \(\hat{\eta}(A)=\lim_{k\to\infty} \sup \eta(big\otimes^k A)^{1/k}\). In this paper, the equivalence of the following conditions are proved: (i) \(\eta(A)<1\), (ii) \(\hat{\eta}(A)<1\), (iii) \(\mu(A)<1\); (iv) \(\lim_{k\to \infty} \otimes^k A=0\), where \(\mu(A)\) is the maximum circuit geometric mean of the directed graph associated with \(A\).