前辅文
1 Introduction
1.1 Kicked oscillators
1.2 Poincar′e 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 Model I
2.3.2 Model II
2.3.3 Model III
2.4 Proof of renormalizability
2.5 Structure of the discontinuity set
2.5.1 Model I
2.5.2 Model III
2.6 More general renormalization
2.7 The p=7 model
References
3 Symbolic Dynamics
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 l
4.4 A complicated example: Model II
4.5 Discontinuity set in Model III
4.6 Multifractal residual set of the p=7 model
4.7 Asymptotic factorization
4.8 Telescoping
4.9 Unique ergodicity for each S (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 p=7 model
References
5 Global Dynamics
5.1 Global expansivity
5.1.1 Lifting the return map rK(0)
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 (0;k;6￥) sectors
5.4.2 Numerical investigations
5.4.3 A non-expansive sector
5.4.4 Generic behavior
References
6 Transport
6.1 Probability calculation using recursive tiling
6.2 Ballistic transport in Model I
6.3 Subdiffusive transport in Model II
6.4 Diffusive transport in Model II
6.5 Superdiffusive transport in Model III
6.6 Discussion
References
7 Hamiltonian Round-Off
7.1 Vector field
7.2 Localization
7.3 Localization of the vector field and periodic orbits
7.4 Symbolic codes for walks
7.5 Construction of the probability distribution
7.6 Rotation number 1/5
7.6.1 Recursive tiling for the local map
7.6.2 Probability distribution P(x; t)
7.6.3 Fractal snowflakes
7.6.4 Substitution rules for lattice walks
7.6.5 Separating out an asymptotic walk
7.6.6 Asymptotic scaling
7.7 Model I
7.8 Model II
7.9 A conjecture
References
Appendix A Data Tables
A.1 Model I Data Tables, from Kouptsov et al. (2002).
A.1.1 Generating domain
A.1.2 Level-0 scaling sequence domains
A.1.3 Level-0 periodic domains
A.1.4 Miscellaneous periodic domains
A.2 Model II Data Tables, from Kouptsov et al. (2002)
A.2.1 Generating partition
A.2.2 Level-0 scaling domains, sequence A
A.2.3 Level-0 periodic domains, sequence A
A.2.4 Miscellaneous periodic domains, j > 10
A.2.5 Level-0 scaling domains, sequence B.
A.2.6 Level-0 periodic domains, sequence B.
A.2.7 Incidence matrices
A.3 Model III Data Tables, from Kouptsov et al. (2002)
A.3.1 Generating domain
A.3.2 Pre scaling level L = ?1
A.3.3 Domains Dj(L) for even L.
A.3.4 Domains Dj(L) for odd L
A.3.5 Domains Pj(L) for all L
A.3.6 Tiling data
A.3.7 Section of the discontinuity set
A.4 Cubic Model Data Tables, from Lowenstein et al. (2004)
A.5 Inadmissibility Tables for Models II and III
References
Appendix B The Codometer
Index
Color Figure Index
