Advanced Series in Nonlinear Dynamics - Vol. 5
COMBINATORIAL DYNAMICS AND ENTROPY IN DIMENSION ONE
2nd Edition
by Lluís Alsedà, Jaume Llibre (Universitat Autònoma de Barcelona) & Michal Misiurewicz (Indiana University)
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.
Contents:
- 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
Readership: Students of applied mathematics and dynamical systems.
Review of 1st Edition
"As a whole, the book is carefully written and contains a very detailed account of a body of material along with some new results. The book will serve as a valuable reference for those interested in the combinatorial aspects of one-dimensional dynamical systems."
| 432pp |
Pub. date: Nov 2000 |
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