准混沌冲击振子：重正化，符号动力学及运动迁移现象（英文版）_（美）洛文斯坦　著_9787040322798

　　《准混沌冲击振子：重正化符号动力学及运动迁移现象（英文版）》介绍了准混沌运动研究的最新进展，讨论了动力系统中有序运动与无序运动交界处的复杂的动力学分支行为。准混沌运动是由具有自相似结构的稳定运动岛邻域附近运动轨迹的吸引性来刻画的，并且其相空间的位移是随时间的幂指数而渐近增加的。本专著全面、系统、自成体系地研究了一维经典冲击振子模型，并以完美的形式展示了准混沌运动在物理学和数学上的规则性和复杂性。
　　《准混沌冲击振子：重正化符号动力学及运动迁移现象（英文版）》包含了目前文献中很多不曾涉及的新内容和新结果，它将激发物理学、应用数学的研究生和学者以及非线性动力学的专家对准混沌运动研究的极大兴趣，是一本难得的教科书或参考书。
　　John H.Lowenstein为纽约大学物理系教授，非线性动力系统领域知名科学家，长期专注于一维冲击振子的动力学行为研究并取得了丰硕的成果，其中包括：在低维混沌和准混沌哈密顿系统中的运动迁移现象，区间及多边形分段等距自相似结构的数学理论。

1 Introduction
1.1 Kicked oscillators
1.2 Poincare sections
1.3 Crystalline symmetry
1.4 Stochastic webs
1.5 Normal and anomalous diffusive behavior
1.6 The sawtooth web map
1.7 Renormalizability
1.8 Long-time asymptotics
1.9 Linking local and global behavior
1.10 Organization of the book
References

2 Renormalizability of the Local Map
2.1 Heuristic approach to renormalizability
2.1.1 Generalized rotations
2.1.2 Natural return map tree
2.1.3 Examples
2.2 Quadratic piecewise isometries
2.2.1 Arithmetic preliminaries
2.2.2 Domains
2.2.3 Geometric transformations on domains
2.2.4 Scaling sequences
2.2.5 Periodic orbits
2.2.6 Recursive tiling
2.2.7 Computer-assisted proofs
2.3 Three quadratic models
2.3.1 Modell
2.3.2 Modelll
2.3.3 Model III
2.4 Proofofrenormalizability
2.5 Structure of the discontinuity set
2.5.1 Modell
2.5.2 Modellll
2.6 More general renormalization
2.7 The π/7 model
References

3 Symbolic Dynanucs
3.1 Symbolic representation of the residual set
3.1.1 Hierarchical symbol strings
3.1.2 Eventually periodic codes
3.1.3 Simplified codes for quadratic models
3.2 Dynamical updating of codes
3.3 Admissibility
3.3.1 Quadratic example
3.3.2 Models I, II, and III
3.3.3 Cubic example
3.4 Minimality
References

4 Dimensions and Measures
4.1 Hausdorff dimension and Hausdorff measure
4.2 Construction of the measure
4.3 Simplification for quadratic irrational
4.4 A complicated example: Model II
4.5 Discontinuity set in Model III
4.6 Multifractal residual set of the π/7 model
4.7 Asymptotic factorization
4.8 Telescoping
4.9 Unique ergodicity for each ∑（i）
4.10 Multifractal spectrum of recurrence time dimensions
4.10.1 Auxiliary measures and dimensions
4.10.2 Simpler calculation of the recurrence time dimensions
4.10.3 Recurrence time spectrum for the π/7 model
References

5 Global Dynanucs
5.1 Global expansivity
5.1.1 Lifting the return map PK （O）
5.1.2 Lifting the higher-level return maps
5.2 Long-time asymptotics
5.3 Quadratic examples
5.4 Cubic examples
5.4.1 Orbits in the （O,k,6∞） sectors
……
6 Transport
7 Hamiltonian Round-Off
Appendix A Data Tables
Appendix B The Codometer
Index
Color Figure Index

　　Chapter 4
　　Here we focus on the measure-theoretic and fractal （ormultifractal） properties of the residual set of the local map. Theintroduction of Hausdorff dimension and Haus- dorff measure will beessential， later on， to formulating transport on interesting setsof Lebesgue measure zero. Using a transfer matrix to linksuccessive levels of the renormalization hierarchy， we calculate anontrivial spectrum of generalized dimensions for the π/7model.
　　Chapter 5
　　We now examine how the localreturn maps of a scaling sequence canbe lifted to the nfinite kicked-oscillator phase space. In somecases the resulting global translation vectors are globallyexpansive， increasing in length geometrically with increasinglevel. In such cases， we prove that the discontinuity-avoidingaperiodic orbits tend to infinity according to a power law. Westudy this phenomenon numerically in a number of examples. In theparticularly interesting case of the π/7 model， we see thatdifferent initial conditions can lead to a variety of asymptoticpower laws.
　　Chapter 6
　　A statistical formulation of the time evolution is presented inthis chapter， based on a coarse-grained probability distributionP（（m，n），t）. Once again， the renormaliz- ability of the models andespecially the recursive tiling property， serve as the organizingprinciples. We consider examples in which the probability issupported on the 2-dimensional periodic domains， but also onesrestricted to interesting lower- dimensional sets. Highly efficientcodometer updating allows us to investigate numerically asymptoticpower laws with log-periodic modulation.
　　Chapter 7
　　In this final chapter， we consider a different type of kickedoscillator dynamics， namely that of a Hamiltonian round-off map（Lowenstein et al， 1997; Lowenstein and Vivaldi， 1998， 2000）. Onceagain we have a local piecewise isometry which induces apermutation of 22. Although in this class of models all orbits areperiodic， one can still describe the long-time behaviorstatistically in terms of periodically and quasi-periodicallymodulated power laws. We study these both analytically and withhighly efficient numerical experiments.
　　Appendixes
　　For the sake of completeness， we include in the Appendix A anumber of data tables for the models which appear throughout thebook.ln addition， we have a short explanation of our implementationof the hierarchical symbolic dynamics， the codometer， in AppendixB.
　　……

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