The max-plus semifield \(\mathbb{R}_{\max}\) is the set \(\mathbb{R}\cup\{-\infty\}\) equipped with the operations \(\boxplus=\max\) and \(\boxtimes= +\). The author investigates the action of semigroups of \(d\times d\) matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.
In terms of orbits, the main result of the paper states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most \(d\)! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton. The proof of the main result is preceded by some elements of spectral and asymptotic theory of max-plus matrices and by the proof of a theorem on nice semigroups of matrices \(S\subset\mathbb{R}^{d\times d}_{\max}\), which implies the main result for semigroups of projective maps defined by a nice semigroup of matrices.
In the final part of the paper, the author deduces as a corollary of the main result the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.