Finite-range topical functions and uniformly topical functions

Abstract

We introduce a remarkable subclass of the class of topical functions, the class of uniformly topical functions, whose dynamical behaviour is investigated. Every uniformly topical endofunction has a spectral vector, related to some special fixed points (possibly at infinity), about which we establish various properties. In the stochastic case, we prove a multiplicative ergodic theorem, asserting that the stochastic spectral vector exists in all cases.

Cycle time of stochastic max-plus linear systems.

“…Actually, it also implies that 1 n x(n, 0) n∈N almost surely if the A(n) have fixed support (that is P(A ij (n) = −∞) ∈ {0, 1}) and the powers of the shift are ergodic, which is an improvement of [1]. It also allows to prove the convergence when the diagonal entries of the A(n) are almost surely finite, under weaker integrability conditions than in [5] (see [17] or [16] for details). 1.…”

“…Bousch and Mairesse proved (Cf. [5]) that, if A(0)0 is integrable, then the sequence 1 n y(n, 0) n∈N converges almost-surely and in mean and that, under stronger integrability conditions, 1 n x(n, 0) n∈N converges almost-surely if and only if the limit of 1 n y(n, 0) n∈N is deterministic. The previous results can be seen as providing sufficient conditions for this to happen.…”

“…Some sufficient conditions for the existence of this cycle time were given by J.E. Cohen [8], F. Baccelli and Liu [4,1], Hong [14] and more recently by Bousch and Mairesse [5], the author [16] or Heidergott et al [13].…”

“…Let L be the limit of 1 n y(n, 0) n∈N , which exists according to [5,Theorem 6.7] and is assumed to be deterministic.…”

“…The stronger integrability ensures the convergence of 1 n x(n, 0) to this limit, like in [5,Theorem 6.18]. There, it appeared as the specialization of a general condition for uniformly topical operators, whereas in this paper it ensures that B0 is integrable for every submatrix B of A(0) with at least one finite entry on each row.…”