Combinatorial Dynamics and Entropy in Dimension One

by Alseda, Lluis; Llibre, Jaume; Misiurewicz, Michal

• ISBN13: 9789810240530
• ISBN10: 9810240538
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-02-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

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.

Table of Contents

Preface	p. xi
Changes in the second edition	p. xvi
Preliminaries	p. 1
General notation	p. 1
Graphs, loops and cycles	p. 5
Interval maps	p. 17
The Sharkovskii Theorem	p. 17
Maps with the prescribed set of periods	p. 32
Forcing relation	p. 42
Patterns for interval maps	p. 44
Antisymmetry of the forcing relation	p. 51
P-monotone maps and oriented patterns	p. 59
Consequences of Theorem 2.6.13	p. 72
Stability of patterns and periods	p. 79
Primary patterns	p. 82
Extensions	p. 85
Characterization of primary oriented patterns	p. 92
More about primary oriented patterns	p. 101
Circle maps	p. 105
Liftings and degree of circle maps	p. 105
Lifted cycles	p. 114
Cycles and lifted cycles	p. 117
Periods for maps of degree different from -1, 0 and 1	p. 120
Periods for maps of degree 0	p. 123
Periods for maps of degree -1	p. 126
Rotation numbers and twist lifted cycles	p. 135
Estimate of a rotation interval	p. 156
Periods for maps of degree 1	p. 162
Maps of degree 1 with the prescribed set of periods	p. 168
Other results	p. 172
Appendix: Lifted patterns	p. 175
Entropy	p. 185
Definitions	p. 185
Entropy for interval maps	p. 196
Horseshoes	p. 203
Entropy of cycles	p. 219
Continuity properties of the entropy	p. 234
Semiconjugacy to a map of a constant slope	p. 249
Entropy for circle maps	p. 261
Proof of Theorem 4.7.3	p. 277
Appendix: Graph maps	p. 291
Combinatorial dynamics	p. 291
Preliminary definitions	p. 291
Star maps	p. 294
Tree maps	p. 306
Graph maps	p. 318
Topological entropy	p. 321
Entropy of monotone models of graph maps	p. 325
Zero entropy patterns for tree maps	p. 329
A characterization of maps with positive topological entropy in terms of the set of periods	p. 331
Lower bounds of the entropy depending on the set of periods	p. 333
Periodic points and topological entropy of transitive graph maps	p. 335
Periods and entropy of transversal graph maps	p. 343
Preliminary definitions	p. 343
Periodic Lefschetz numbers	p. 344
Periods of transversal graph maps	p. 345
Entropy of continuous graph maps	p. 346
Appendix: Rotation Theory	p. 349
General setup	p. 349
Subshifts of finite type	p. 354
Interval maps	p. 358
Other systems	p. 362
Bibliography	p. 365
List of symbols	p. 403
Index	p. 405
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