On matrix polynomials and the joint spectral radius over max-algebras

Abstract

Our aim is to study matrix polynomials over max-algebras and their growth in terms of max-induced seminorms. In particular, we compare the set growth of a bounded family $\Psi$ of matrix polynomials, measured in terms of the seminorms ${\eta }_{\Vert \cdot \Vert }$ and ${\hat{\eta }}_{\Vert \cdot \Vert }$ with the induced joint spectral radius of the coefficient pool ${\Psi }_0$ of the matrix polynomials. Dynamics of max-linear maps and convergence to periodic points under a single joint spectral radius condition and the existence of common max-eigenvectors of the coefficient pool are also brought out.