On the asymptotic stability of nonnegative matrices in max algebra

In the max algebra system, for an n × n nonnegative matrix A = [aij] the eigenequation for max eigenvalue λ and corresponding max eigenvector x is A ⊗ x = λx, where [A ⊗ x]i = max1⩽j⩽naijxj and μ(A) is the maximum circuit geometric mean. It is shown that the following conditions are mutually equivalent: (i) η∥·∥(A) < 1, for some norm ∥·∥ on Rn; (ii) ηˆ(A)<1; (iii) μ(A) < 1; (iv) limk→∞A⊗k=0, where η∥·∥(A) = max∥x∥ = 1, x⩾0∥A ⊗ x∥ and ηˆ(A)=limsupk→∞[η‖·‖(A⊗k)]1k.