by Lluís Alsedà (Universitat Autònoma de Barcelona) , Jaume Llibre (Universitat Autònoma de Barcelona) , & Michal Misiurewicz ( 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.

• 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