## Complex SystemsThis volume contains the courses given at the Sixth Summer School on Complex Systems held at Facultad de Ciencias Fisicas y Maternaticas, Universidad de Chile at Santiago, Chile, from 14th to 18th December 1998. This school was addressed to graduate students and researchers working on areas related with recent trends in Complex Systems, including dynamical systems, cellular automata, complexity and cutoff in Markov chains. Each contribution is devoted to one of these subjects. In some cases they are structured as surveys, presenting at the same time an original point of view and showing mostly new results. The paper of Pierre Arnoux investigates the relation between low complex systems and chaotic systems, showing that they can be put into relation by some re normalization operations. The case of quasi-crystals is fully studied, in particular the Sturmian quasi-crystals. The paper of Franco Bagnoli and Raul Rechtman establishes relations be tween Lyapunov exponents and synchronization processes in cellular automata. The principal goal is to associate tools, usually used in physical problems, to an important problem in cellularautomata and computer science, the synchronization problem. The paper of Jacques Demongeot and colleagues gives a presentation of at tractors of dynamical systems appearing in biological situations. For instance, the relation between positive or negative loops and regulation systems. |

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### Contents

A WORKED OUT EXAMPLE | 1 |

LYAPUNOV EXPONENTS AND SYNCHRONIZATION OF CELLULAR AUTOMATA | 69 |

DYNAMICAL SYSTEMS AND BIOLOGICAL REGULATIONS | 105 |

CELLULAR AUTOMATA AND ARTIFICIAL LIFE | 151 |

WHY KOLMOGOROV COMPLEXITY? | 201 |

SOME EXAMPLES AND APPLICATIONS | 261 |

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### Common terms and phrases

algorithm alphabet attractor biinfinite binary words called cell Cellular Automata cellular space chaotic circuit combinatorics command complexity computation computation-universal condition configuration consider continued fraction coordinates coupling operation cutoff cutting line defined definition Demongeot denote dimension distribution dynamical systems ECAn element entropy function entropy version equivalent example exists Figure finite fixed formula fundamental domain geodesic flow hence infinite information space initial input integer interval exchange Kolmogorov complexity Kolmogorov entropy language lattice Lebesgue measure linear Loop Lyapunov exponents Markov chain Markov partition Math matrix measure notation notion object obtain output pair partition Phys positive probability Proof Proposition random variable Rauzy rectangle renormalization resp return map reversible rotation RPCA S-gate sample chain scenery flow self-reproducing semi-orbit sequence simulate sturmian quasicrystal sturmian word subset subshift symbolic dynamics synchronization Theorem theory tiles tion torus Turing machine vector