Combinatorial dynamics and entropy in dimension one

著者

• Alsedà, Ll
• Llibre, Jaume
• Misiurewicz, Michał

書誌事項

Combinatorial dynamics and entropy in dimension one

Lluís Alsedà & Jaume Llibre, Michał Misiurewicz

（Advanced series in nonlinear dynamics, v. 5）

World Scientific, c2000

2nd. ed

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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詳細情報

• NII書誌ID(NCID): BA49431601
• ISBN:
  • 9810240538
• 出版国コード: si
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Singapore
• ページ数/冊数: xvi, 415 p.
• 大きさ: 23 cm
• 親書誌ID:
  • BA12210467