## Aspects of Ergodic, Qualitative and Statistical Theory of MotionIntended for beginners in ergodic theory, this book addresses students as well as researchers in mathematical physics. The main novelty is the systematic treatment of characteristic problems in ergodic theory by a unified method in terms of convergent power series and renormalization group methods, in particular. Basic concepts of ergodicity, like Gibbs states, are developed and applied to, e.g., Asonov systems or KAM Theory. Many examples illustrate the ideas and, in addition, a substantial number of interesting topics are treated in the form of guided problems. |

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### Contents

General Qualitative Properties | 1 |

12 Examples and Some Definitions | 3 |

13 Harmonic Oscillators and Integrable Systems as Dynamical Systems | 13 |

14 Frequencies of Visit | 17 |

Ergodicity and Ergodic Points | 27 |

22 Ergodic Properties of QuasiPeriodic Motions Ergodic Sequences and Measures | 32 |

23 Ergodic Points | 52 |

24 Ergodic Decomposition | 62 |

72 Cluster Expansions | 240 |

73 Renormalization by Decimation in OneDimensional Systems | 257 |

More Criteria | 270 |

75 Phase Transitions | 276 |

Special Ergodic Theory Problems in Nonchaotic Dynamics | 291 |

82 Graphs and Diagrams for the Lindstedt Series | 299 |

83 Cancellations | 307 |

84 Convergence and the KAM Theorem | 316 |

Entropy and Complexity | 73 |

32 The ShannonMcMillan Theorem | 81 |

33 Elementary Properties of the Average Entropy | 92 |

34 Further Properties of the Average Entropy Generator Theorem | 99 |

Markovian Pavements | 109 |

42 Markovian Pavements for Hyperbolic Systems | 119 |

43 Coding of the Volume Measure of Smooth Hyperbolic Systems | 141 |

Gibbs Distributions | 155 |

52 Properties of Gibbs Distributions | 165 |

53 Gibbs Distributions on Z+ | 170 |

Expansive Maps of 01 | 180 |

General Properties of Gibbs and SRB Distributions | 189 |

62 Applications to Anosov Systems SRB Distribution | 199 |

63 Periodic Orbits Invariant Probability Distributions and Entropy | 208 |

64 Equivalent Potentials Gibbs Distributions with Transitive Vacuum | 216 |

Analyticity Singularity and Phase Transitions | 227 |

Some Special Topics in KAM Theory | 327 |

92 Bounds on Renormalized Series Convergence | 332 |

93 Scaling Laws for the Standard Map | 341 |

94 Limit Function for the Standard Map near Resonances | 352 |

Special Problems in Chaotic Dynamics | 359 |

102 Extended Systems Lattices of Arnolds Cat Maps | 366 |

An SRB Distribution | 371 |

104 Chaos in SpaceTime and SRB Distributions | 381 |

105 Isomorphisms | 392 |

A Nonequilibrium Thermodynamics? TwentySeven Comments | 397 |

Bibliography | 411 |

423 | |

425 | |

433 | |

### Other editions - View all

Aspects of Ergodic, Qualitative and Statistical Theory of Motion Giovanni Gallavotti,Federico Bonetto,Guido Gentile No preview available - 2014 |

Aspects of Ergodic, Qualitative and Statistical Theory of Motion Giovanni Gallavotti,Federico Bonetto,Guido Gentile No preview available - 2010 |

### Common terms and phrases

analytic Anosov apply average Borel bounded branches called close compatibility complex condition connected consider constant construction contains convergence corollary corresponding defined definition denote depend derivative discussed dynamical system elements entropy equation ergodic example exists expansion expression fact factor finite fixed function Furthermore Gibbs distribution given hence Holder continuous holds implies independent integrable interesting interval invariant labels latter lattice length limit manifold Markovian matrix means measure metric mixing motions natural nodes Note obtained partition pavement periodic phase Physics points possible potential probability distribution problem proof proposition prove radius of convergence relation Remark replaced respect result rotation satisfies scale Sect self-energy graphs sequence Show side space suitable suppose symbolic theorem theory topological transitive tree unique unstable values vector