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

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 $\mathrm{\Psi }$ of matrix polynomials, measured in terms of the seminorms ${\eta }_{\mathrm{\parallel }\cdot \mathrm{\parallel }}$ and ${\widehat{\eta }}_{\mathrm{\parallel }\cdot \mathrm{\parallel }}$ with the induced joint spectral radius of the coefficient pool ${\mathrm{\Psi }}_0$ 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.