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## Series Expansions of Lyapunov Exponents and (2000)

### Citations

3132 | Perturbation Theory for Linear Operators - Kato - 1976 |

384 |
Nonnegative matrices in the mathematical sciences
- Berman, Plemmons
- 1994
(Show Context)
Citation Context ... observation. Recall that the Perron root of a nonnegative matrix is by definition its spectral radius, which is an eigenvalue associated to a nonnegative eigenvector, by the Perron-Frobenius theorem =-=[6]-=-. Lemma 3. If P is a (nonnegative) trim linear representation of a rational probability measure p, then, the Perron root of the matrix P is equal to 1 and is semisimple. Besides, the eigen... |

356 | Regular Algebra and Finite Machines. - Conway - 1971 |

287 | Transductions and Context-Free Languages.
- Berstel
- 1979
(Show Context)
Citation Context ..., i 1 flfiffififfifi k, we rewrite (19) as: ak fiffififfifi a1x zk k i 1s ai zi ' 1 ,fi (20) (When S is finite, a cocycle representation is exactly a subsequential transducer =-=[7]-=- with output in the monoid +* .) If a1 a2 flfiffififfifi is a random sequence of independent identically distributed elements INRIA Series Expansions of Lyapunov Exponents and Forgetful Mono... |

286 |
D/Perrin, Theory of codes,
- Berstel
- 1985
(Show Context)
Citation Context ... p z s r fi (27) We shall first derive (26) and (27) from results on codes, and then, we shall give a second, probably more intuitive, probabilistic proof. Let us recall some definitions from =-=[8]-=- (the reader should consult this book for more details, and, in this proof, all references are relative to this source). A subset X is a code if it generates a free monoid, and it is prefix if f... |

201 | Rational series and their languages - Berstel, Reutenauer - 1988 |

188 |
Semigroups and Combinatorial Applications
- Lallement
- 1979
(Show Context)
Citation Context ... ac ' 1 B ) ac (45) RR n° 3971 22 Stéphane Gaubert , Dohy Hong where B a , and where, as usual, we write ac instead of ac . Since the rational expression in (45) is unambiguous =-=[35]-=-, we obtain Zr def r p by replacing by , concatenation by product, and % by 1 % ' 1 in (45). After simplification: Zr p a c 1 b a p b 1 p ... |

183 |
Products of Random Matrices with Applications to Schrödinger operators
- Bougerol, Lacroix
- 1985
(Show Context)
Citation Context ...ndition ensures that f d d ). If each map fk is of the form fk x log Mk exp x , for some Mk , then the Lyapunov exponent (1) coincides with the classical top Lyapunov exponent =-=[11]-=- of the random product of nonnegative matrices Mn fiffififfifi M1, which is defined by: a fi s fi lim n ! " 1 n log Mn %ffi%ffi% M1 for any norm % . The second and main example of monotone... |

147 |
Noncommuting random products.
- Furstenberg
- 1963
(Show Context)
Citation Context ...on the set of states reachable from x by the action of . 2.4 Furstenberg’s Cocycle Formula We next specialize to our discrete context Furstenberg’s cocycle representation of the Lyapunov exponent =-=[21]-=-. Given an action of on a denumerable set S, S S y % y, we say that a maps: S is a cocycle ifs u y s u % y s y holds for all ysS and u s... |

93 | Methods and applications of (max, +) linear algebra. In:
- Gaubert, Plus
- 1997
(Show Context)
Citation Context ...d if its critical graph has a single strongly connected component with cyclicity 1 — there is a power of a whose image is a line (the max-plus spectral theorem has been proved by various authors, see =-=[1, 39, 27, 5]-=- for recent references). When ac , it is easy to see that r a flfiffififfifi ac ' 1 B ) ac , B ) a flfiffififfifi ac ' 1 B ) ac (45) RR n° 3971 22 Stéphane Gaub... |

89 |
Some relations between nonexpansive and order preserving mappings
- Crandall, Tartar
- 1980
(Show Context)
Citation Context ...badditive (i.e. if Sn ) k Sn Sk), then the limit limn Sn n exists. This shows thatsexists when x 0. To show thatsexists for any x , it suffices to use the following classical easy observation =-=[18]-=-: a monotone homogeneous map f is non-expansive for the sup norm % , i.e. f y sf y ysy , for all y y s d . Hence, if Sn y and Sn y denote the sums (8) evaluated ... |

56 | Modeling and analysis of timed Petri nets using heaps of pieces.
- Gaubert, Mairesse
- 1999
(Show Context)
Citation Context ...hows the heap of pieces corresponding to the sequence: a b a c b b b a. Here, n 8 and Xn 4 6 . It is quite easy to see that Xn is given by the random dynamical system (2) (see =-=[24, 12, 26]-=- for details). In this context, the Lyapunov exponent is equal to the almost sure limit of the height of a heap of pieces, divided by the number of pieces, when the number of pieces grows to infinity.... |

53 | Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes - Cartan - 1978 |

45 | A survey of Matrix theory and matrix inequalities. Allyn and - MARCUS, MINe - 1964 |

40 | Performance Evaluation of (max,+) Automata.
- Gaubert
- 1995
(Show Context)
Citation Context ...unov exponent of the heap model of Fig. 1. Computing exactly, or approximating, Lyapunov exponents of heaps of pieces, and more generally, of products of max-plus matrices, is a long standing problem =-=[15, 44, 41, 1, 28, 23, 12, 3, 4, 25, 19, 10]-=-. No exact formulæ are known, except in very special cases, such as the one of Fig. 2. In this paper, our purpose is rather to investigate the qualitative properties ofs. For instance, a simple look a... |

33 |
Idempotent analysis, volume 13
- Maslov, Samborskiĭ
- 1992
(Show Context)
Citation Context ...d if its critical graph has a single strongly connected component with cyclicity 1 — there is a power of a whose image is a line (the max-plus spectral theorem has been proved by various authors, see =-=[1, 39, 27, 5]-=- for recent references). When ac , it is easy to see that r a flfiffififfifi ac ' 1 B ) ac , B ) a flfiffififfifi ac ' 1 B ) ac (45) RR n° 3971 22 Stéphane Gaub... |

32 |
A max version of the Perron–Frobenius theorem, Linear Algebra Appl.
- Bapat
- 1998
(Show Context)
Citation Context ...d if its critical graph has a single strongly connected component with cyclicity 1 — there is a power of a whose image is a line (the max-plus spectral theorem has been proved by various authors, see =-=[1, 39, 27, 5]-=- for recent references). When ac , it is easy to see that r a flfiffififfifi ac ' 1 B ) ac , B ) a flfiffififfifi ac ' 1 B ) ac (45) RR n° 3971 22 Stéphane Gaub... |

30 | On the existence of cycle times for some nonexpansive maps
- Gunawardena, Keane
- 1995
(Show Context)
Citation Context ... . Then, # Xn # Xn i need not be subadditive or superadditive, and even in the case of a deterministic sequence ( f1 f2 fiffififfifi ), a counter-example, due to Gunawardena and Keane =-=[29]-=- (see Remark 1 below), shows that the -Lyapunov exponent, limn Xn i n, need not exist (however, Theorem 2 below shows that under additional assumptions, the -Lyapunov exponent does exist). T... |

30 | Products of irreducible random matrices in the (max,+) algebra
- Mairesse
- 1997
(Show Context)
Citation Context ...n in our Theorem 2 is the memory loss property, whose importance, in the context of heaps of pieces, or more generally, of products of max-plus random matrices, has been recognized by several authors =-=[37, 24, 12, 19]-=-. For the heap model of Fig. 1, this assumption just means that the arrival of a rigid piece (piece c) occupying all the slots, resets the heap to a state identical, up to a vertical translation, to t... |

29 | Subadditivity, Generalized Products of Random Matrices and - Cohen - 1988 |

28 |
Some ergodic results on stochastic iterative discrete events systems
- Vincent
- 1997
(Show Context)
Citation Context ...n $%ffi%ffi%&$ f1 x ] fi (1) Thus, the top Lyapunov exponent measures the linear growth rate of the orbits of the random dynamical system: Xn fn Xn ' 1 X0 x fi (2) As observed by Vincent =-=[47]-=- (see §2 below for details), the limit in (1), if it exists, is independent of x , and, when t f1 0 is integrable, the existence of the limit follows from the fact that the sequence Sn # [t fn (... |

26 |
Domains of analytic continuation for the top lyapunov exponent. Annales de l’Institut Henri Poincare. Probabilites et Statistiques,
- Peres
- 1992
(Show Context)
Citation Context ... obtain a way to approximate Lyapunov exponents. In general, the Lyapunov exponent need not be differentiable (look at the point p a p b 1 2 in Fig. 2), and it may even be discontinuous =-=[43]-=-. The critical assumption in our Theorem 2 is the memory loss property, whose importance, in the context of heaps of pieces, or more generally, of products of max-plus random matrices, has been recogn... |

25 |
Limit theorems for products of positive randommatrices.Ann
- Hennion
- 1997
(Show Context)
Citation Context ...more accurate estimation of the analyticity domain. However, the memory loss property remains in essence, similar to the contraction properties in the projective space, used by Peres, and many others =-=[11, 36, 32]-=- (memory loss is indeed a very strong “ultimate” contraction property, of Lipschitz constant 0). It would be very interesting to prove similar results without contraction arguments. For instance, we d... |

24 |
Discrete event systems with stochastic processing times.
- Olsder, Resing, et al.
- 1990
(Show Context)
Citation Context ...unov exponent of the heap model of Fig. 1. Computing exactly, or approximating, Lyapunov exponents of heaps of pieces, and more generally, of products of max-plus matrices, is a long standing problem =-=[15, 44, 41, 1, 28, 23, 12, 3, 4, 25, 19, 10]-=-. No exact formulæ are known, except in very special cases, such as the one of Fig. 2. In this paper, our purpose is rather to investigate the qualitative properties ofs. For instance, a simple look a... |

17 | Approximating the spectral radius of sets of matrices in the max-algebra is NP-hard
- Blondel, Tsitsiklis
(Show Context)
Citation Context ...unov exponent of the heap model of Fig. 1. Computing exactly, or approximating, Lyapunov exponents of heaps of pieces, and more generally, of products of max-plus matrices, is a long standing problem =-=[15, 44, 41, 1, 28, 23, 12, 3, 4, 25, 19, 10]-=-. No exact formulæ are known, except in very special cases, such as the one of Fig. 2. In this paper, our purpose is rather to investigate the qualitative properties ofs. For instance, a simple look a... |

14 | Report on the program AMoRE
- Matz, Miller, et al.
- 1995
(Show Context)
Citation Context ... of the other cases, it is easy to write an unambiguous rational expression for t or r , from which the absolute convergence domain of Z can be obtained. We checked these computations using AMoRE =-=[40]-=- and MAPLE. -1.5 -1 -0.5 0 0.5 1 1.5 y -1 -0.5 0 0.5 1 1.5 2 x -1.5 -1 -0.5 0 0.5 1 1.5 y -1 -0.5 0 0.5 1 1.5 2 x a2 a2 b2 z 1 1 z 1 1 z z 1 -1.5 -1 -0.5 0 ... |

14 |
Easy multiplications I. The Realm of Kleene’s Theorem
- Sakarovitch
- 1987
(Show Context)
Citation Context ...ctor u makes the product forget its right factor, which accounts for the name of the monoid). When the set is rational, the associated forgetful monoid is nothing but a very special rational monoid =-=[45, 42]-=-. We obtain in passing an Abelian representation of the Lyapunov exponent (Theorem 1), which plays for Lyapunov exponents, mutatis mutandis, the role that resolvents plays for eigenvalues of linear ma... |

13 |
Dynamics of synchronized parallel systems
- Brilman, Vincent
- 1997
(Show Context)
Citation Context ...ry. If each map fk is of the form (5) for some matrix Mk , the Lyapunov exponent (1) coincides with the Lyapunov exponent of the random product of matrices Mn fiffififfifi M1 in the max-plus semiring =-=[1, 15, 12]-=-. An appealing example of max-plus random products is provided by Tetris-like heaps of pieces. For instance, consider the three monotone homogeneous maps 2 2 : a x x1 1 x2 T b ... |

10 |
Concurrency measure in commutation monoids
- Saheb
- 1989
(Show Context)
Citation Context |

9 |
Analytic expansions of max-plus Lyapunov exponents
- Baccelli, Hong
(Show Context)
Citation Context ...s of products of random matrices. A classical one is the monograph [11]. A recent one, dedicated to the case of nonnegative matrices, is [32]. Some of the results of this paper have been announced in =-=[2]-=-. 2 Probability Measures on Words and Lyapunov Exponents 2.1 Notation and Definitions Given a finite alphabet , we denote by k the set of words of length k, i.e. the set of sequences of the form ... |

9 |
Lyapunov exponents of large, sparse random matrices and the problem of directed polymers with complex random weights
- Cook, Derrida
- 1990
(Show Context)
Citation Context ...s a function of the horizon, when the transition costs or rewards are random [15]. Max-plus Lyapunov exponents also arise in Statistical Physics, in the study of disordered systems at low temperature =-=[22, 17]-=-. There is a number of contributions on Lyapunov exponents of products of random matrices. A classical one is the monograph [11]. A recent one, dedicated to the case of nonnegative matrices, is [32]. ... |

9 |
An elementary proof of the Birkhoff-Hopf theorem
- Eveson, Nussbaum
- 1995
(Show Context)
Citation Context ...the assumption “there exists an iterate that is strictly contracting for Hilbert’s projective metric” by “the image of one iterate is a compact set in the projective space” (the Birkhoff-Hopf theorem =-=[20]-=- shows that both statements are equivalent in the special case of linear maps acting in the positive cone, hence, this question is only interesting for monotone homogeneous maps which are not linear i... |

8 |
Analyticity of iterates of random non-expansive maps
- Baccelli, Hong
- 2000
(Show Context)
Citation Context |

8 |
Mesures de probabilites rationnelles.
- Hansel, Perrin
- 1990
(Show Context)
Citation Context ...flfiffififfifi ak b1 flfiffififfifi bls . The free monoid is obtained by adjoining to ) the empty sequence, which is called the empty word in this context. Following Hansel and Perrin =-=[30]-=-, we say that a map p : [0 1] is a probability measure on words if a p a p , for all s , and if p 1 1. This implies that k p 1, for all k. We say ... |

6 | Asymptotic analysis of heaps of pieces and application to timed Petri nets.
- Gaubert, Mairesse
- 1999
(Show Context)
Citation Context |

6 |
Répartition d’état d’un opérateur de Schrödinger aléatoire
- Page
- 1984
(Show Context)
Citation Context ...more accurate estimation of the analyticity domain. However, the memory loss property remains in essence, similar to the contraction properties in the projective space, used by Peres, and many others =-=[11, 36, 32]-=- (memory loss is indeed a very strong “ultimate” contraction property, of Lipschitz constant 0). It would be very interesting to prove similar results without contraction arguments. For instance, we d... |

5 |
generalized products of random matrices and
- Subadditivity
- 1988
(Show Context)
Citation Context ...ry. If each map fk is of the form (5) for some matrix Mk , the Lyapunov exponent (1) coincides with the Lyapunov exponent of the random product of matrices Mn fiffififfifi M1 in the max-plus semiring =-=[1, 15, 12]-=-. An appealing example of max-plus random products is provided by Tetris-like heaps of pieces. For instance, consider the three monotone homogeneous maps 2 2 : a x x1 1 x2 T b ... |

5 |
Computational issues in stochastic recursive systems
- Gaujal, Marie
- 1998
(Show Context)
Citation Context |

4 |
On the maximal throughput of a synchronization process. Submitted to Mathematics of Operation research
- Robert
- 1998
(Show Context)
Citation Context |

4 | Zero temperature magnetization of a one-dimensional spin glass
- Gardner, Derrida
- 1985
(Show Context)
Citation Context ...s a function of the horizon, when the transition costs or rewards are random [15]. Max-plus Lyapunov exponents also arise in Statistical Physics, in the study of disordered systems at low temperature =-=[22, 17]-=-. There is a number of contributions on Lyapunov exponents of products of random matrices. A classical one is the monograph [11]. A recent one, dedicated to the case of nonnegative matrices, is [32]. ... |

3 |
Task resource systems and
- Gaubert, Mairesse
- 1995
(Show Context)
Citation Context ...hows the heap of pieces corresponding to the sequence: a b a c b b b a. Here, n 8 and Xn 4 6 . It is quite easy to see that Xn is given by the random dynamical system (2) (see =-=[24, 12, 26]-=- for details). In this context, the Lyapunov exponent is equal to the almost sure limit of the height of a heap of pieces, divided by the number of pieces, when the number of pieces grows to infinity.... |

2 |
Easy multiplications ii. extentions of rational semigroups
- Pelletier, Sakarovitch
- 1990
(Show Context)
Citation Context ...ctor u makes the product forget its right factor, which accounts for the name of the monoid). When the set is rational, the associated forgetful monoid is nothing but a very special rational monoid =-=[45, 42]-=-. We obtain in passing an Abelian representation of the Lyapunov exponent (Theorem 1), which plays for Lyapunov exponents, mutatis mutandis, the role that resolvents plays for eigenvalues of linear ma... |