1 Introduction
1.1 Kicked oscillators
1.2 Poincaré 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 ModelⅠ
2.3.2 ModelⅡ
2.3.3 Model Ⅲ
2.4 Proof of renormalizability
2.5 Structure of the discontinuity set
2.5.1 ModelⅠ
2.5.2 ModelⅢ
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 λ
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 Dynamics
5.1 Global expansivity
5.1.1 Lifting the return map pK （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 transportin ModelⅡ
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 ModelⅠ
7.8 Model Ⅱ
7.9 A conjecture
References
Appendix AData Tables
A.1 Modell 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 ModelⅡ 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 ModelⅢ 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 Пj（L） for all L
A.3.6 Tiling data
A.3.7 Section of the discontinuity set
A.4 Cubic ModelData Tables, from Lowenstein et al. （2004）
A.5 Inadmissibility Tables for Models Ⅱ and Ⅲ
References
Appendix B The Codometer
Index
Color Figure Index
摘要
It is not immediately obvious that choosing λ to be a low-degree algebraic inte-ger should help our search for dynamical self-similarity （beyond the restriction that it places on the denominator of the rotation number）. Of course, it is well known that the lowest-degree algebraic integers, solutions of quadratic equations, enjoy algebraic self-similarity in their continued.fraction expansions. Moreover, for one-dimensional maps analogous to piecewise isometries, namely the interval exchange transformations, one has a powerful theorem of Boshernitzan and Carroll （1997） es-tablishing their renormalizability for quadratic irrational parameters. Unfortunately,no comparable theorem for two-dimensional PWI's has been proved. However, for two-dimensional PWI's, the renormalizability of an important class of models with quadratic irrational λ has been rigorously established by Kouptsov et al. （2002） us-ing computer assisted proofs. It is here that the true advantage of the restriction to low-degree algebraic numbers makes itself felt: it makes it possible to use com-puter software to perform exact calculations on specific models, most of which have exceedingly complicated multi-level return map structures, thereby verifying impor-tant properties of each model and, by exhaustion, the entire class. Before examining three particularly interesting models from the class of PWI's of the square with rational rotation numbers and quadratic irrational parameters, it will be useful to illustrate how the systematic search for renormalizable return map structure succeeds in a particularly simple example. The contrast with the λ = 1/2 case will be striking.
