Combinatorial dynamics and entropy in dimension one
著者
書誌事項
Combinatorial dynamics and entropy in dimension one
(Advanced series in nonlinear dynamics, v. 5)
World Scientific, c2000
2nd. ed
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注記
Includes bibliography (p. 365-401) and index
内容説明・目次
内容説明
This book introduces the reader to the two main directions of one-dimensional dynamics. The first has its roots in the Sharkovskii theorem, which describes the possible sets of periods of all cycles (periodic orbits) of a continuous map of an interval into itself. The whole theory, which was developed based on this theorem, deals mainly with combinatorial objects, permutations, graphs, etc.; it is called combinatorial dynamics. The second direction has its main objective in measuring the complexity of a system, or the degree of “chaos” present in it; for that the topological entropy is used. The book analyzes the combinatorial dynamics and topological entropy for the continuous maps of either an interval or the circle into itself.
目次
- Preliminaries: general notation
- graphs, loops and cycles. Interval maps: the Sharkovskii Theorem
- maps with the prescribed set of periods
- forcing relation
- patterns for interval maps
- antisymmetry of the forcing relation
- P-monotone maps and oriented patterns
- consequences of Theorem 2.6.13
- stability of patterns and periods
- primary patterns
- extensions
- characterization of primary oriented patterns
- more about primary oriented patterns. Circle maps: liftings and degree of circle maps
- lifted cycles
- cycles and lifted cycles
- periods for maps of degree different from -1, 0 and 1
- periods for maps of degree 0
- periods for maps of degree -1
- rotation numbers and twist lifted cycles
- estimate of a rotation interval
- periods for maps of degree 1
- maps of degree 1 with the prescribed set of periods
- other results. Appendix: lifted patterns. Entropy: definitions
- entropy for interval maps
- horseshoes
- entropy of cycles
- continuity properties of the entropy
- semiconjugacy to a map of a constant slope
- entropy for circle maps
- proof of Theorem 4.7.3.
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