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.
Combinatorial Dynamics And Entropy In Dimension One (2nd Edition)
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.