Combinatorial Dynamics And Entropy In Dimension One (2nd Edition)
World Scientific Publishing Co Pte Ltd (Verlag)
978-981-02-4053-0 (ISBN)
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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.
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.
| Erscheint lt. Verlag | 1.11.2000 |
|---|---|
| Reihe/Serie | Advanced Series in Nonlinear Dynamics ; 5 |
| Verlagsort | Singapore |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Angewandte Mathematik |
| Mathematik / Informatik ► Mathematik ► Graphentheorie | |
| ISBN-10 | 981-02-4053-8 / 9810240538 |
| ISBN-13 | 978-981-02-4053-0 / 9789810240530 |
| Zustand | Neuware |
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