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28
Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture
 J. of the American Mathematical Society
, 2001
"... A topical map is a map from Rn into itself verifying some conditions (see §1.2) and which, roughly speaking, behaves like a translation along some line, the amount of which is measured by a real number, called the average height (or average displacement) of the map. Then we look at a topical Iterate ..."
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Cited by 71 (5 self)
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A topical map is a map from Rn into itself verifying some conditions (see §1.2) and which, roughly speaking, behaves like a translation along some line, the amount of which is measured by a real number, called the average height (or average displacement) of the map. Then we look at a topical Iterated Function System (IFS),
A constructive fixed point theorem for minmax functions
 DYNAMICS AND STABILITY OF SYSTEMS
, 1999
"... Minmax functions, F: Rn → Rn, arise in modelling the dynamic behaviour of discrete event systems. They form a dense subset of those functions which are homogeneous, Fi(x1 + h, · · · , xn + h) = Fi(x1, · · · , xn) + h, monotonic, ⃗x ≤ ⃗y ⇒ F (⃗x) ≤ F (⃗y), and nonexpansive in the ℓ ∞ norm—so ..."
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Cited by 42 (12 self)
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Minmax functions, F: Rn → Rn, arise in modelling the dynamic behaviour of discrete event systems. They form a dense subset of those functions which are homogeneous, Fi(x1 + h, · · · , xn + h) = Fi(x1, · · · , xn) + h, monotonic, ⃗x ≤ ⃗y ⇒ F (⃗x) ≤ F (⃗y), and nonexpansive in the ℓ ∞ norm—socalled topical functions—which have appeared recently in the work of several authors. Our main result characterises those minmax functions which have a (generalised) fixed point, where Fi(⃗x) = xi + h for some h ∈ R. We deduce several earlier fixed point results. The proof is inspired by Howard’s policy improvement scheme in optimal control and yields an algorithm for finding a fixed point, which appears efficient in an important special case. An extended introduction sets the context for this paper in recent work on the dynamics of topical functions.
Tropical polyhedra are equivalent to mean payoff games
 INT. J. OF ALGEBRA AND COMPUTATION, EPRINT
, 2011
"... ..."
A spectral theorem for convex monotone homogeneous maps
 In Proceedings of the Satellite Workshop on MaxPlus Algebras, IFAC SSSC’01
, 2001
"... Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point ..."
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Cited by 27 (15 self)
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Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex infsubsemilattice of R n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group
Analytic Expansions of (max,+) Lyapunov Exponents
, 1998
"... We give an explicit analytic series expansion of the (max; +)Lyapunov exponent fl(p) of a sequence of independent and identically distributed random matrices in this algebra, generated via a Bernoulli scheme depending on a small parameter p. A key assumption is that one of the matrices has a unique ..."
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Cited by 10 (1 self)
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We give an explicit analytic series expansion of the (max; +)Lyapunov exponent fl(p) of a sequence of independent and identically distributed random matrices in this algebra, generated via a Bernoulli scheme depending on a small parameter p. A key assumption is that one of the matrices has a unique eigenvector. This allows us to use a representation of this exponent as the mean value of a certain random variable, and then a discrete analogue of the socalled lighttraffic perturbation formulas to derive the expansion. We show that it is analytic under a simple condition on p. This also provides a closed form expression for all derivatives of fl(p) at p = 0 and approximations of fl(p) of any order, together with an error estimate for nite order Taylor approximations. Several extensions of this are discussed, including expansions of multinomial schemes depending on small parameters (p 1, ..., p m ) and expansions for exponents associated with iterates of a class of random operators...
Series Expansions of Lyapunov Exponents and Forgetful Monoids
, 2000
"... We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for ..."
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Cited by 5 (0 self)
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We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for Tetrislike heaps of pieces models, we give a series expansion formula for the Lyapunov exponent, as a function of the probability law. In the case of rational probability laws, we show that the Lyapunov exponent is an analytic function of the parameters of the law, in a domain that contains the absolute convergence domain of a partition function associated to a special "forgetful" monoid, defined by generators and relations.
Structure Properties of MinMax Systems and Existence of Global Cycle Time
"... This paper studies minmax systems which are dynamic systems including three operations (min,max ,+) with unknown or stochastic parameters. Some sucient conditions will be given for the existence of global cycle times. Our results are based on structure properties of minmax systems. ..."
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Cited by 5 (3 self)
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This paper studies minmax systems which are dynamic systems including three operations (min,max ,+) with unknown or stochastic parameters. Some sucient conditions will be given for the existence of global cycle times. Our results are based on structure properties of minmax systems.
Cycle time of stochastic maxplus linear systems
 Electronic Journal of Probability
, 2008
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Monotone Rational Series And MaxPlus Algebraic Models Of RealTime Systems
 in &quot;Proc. of the Fourth Workshop on Discrete Event Systems (WODES98
, 1998
"... In the modelling of timed discrete event systems, one traditionally uses dater functions, which give completion times, as a function of numbers of events. Dater functions are nondecreasing. We extend this modelling to the case of multiform logical and physical times, which are needed to model concu ..."
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In the modelling of timed discrete event systems, one traditionally uses dater functions, which give completion times, as a function of numbers of events. Dater functions are nondecreasing. We extend this modelling to the case of multiform logical and physical times, which are needed to model concurrent behaviors. We represent event sequences and time instants by words. A dater is a map, which associates to a word a word, or a set of words, and which is nondecreasing for the subword order. The formal series associated with these generalized dater functions live in a finitely presented semiring, which is equipped with some remarkable relations, due to the monotone character of daters. The implementation of this semiring relies on a theory of rational and recognizable series whose coefficients form a nondecreasing sequence in an idempotent semiring, that we sketch. Finally, we apply this formalism to the modelling and analysis of an elementary example of real time system. Keywords M...