Combinatorial Dynamics And Entropy In Dimension One
Will be published approx. on 1. June 1993
Book
Hardback
344 pages
978-981-02-1344-2 (ISBN)
Description
In last thirty years an explosion of interest in the study of nonlinear dynamical systems occured. The theory of one-dimensional dynamical systems has grown out in many directions. One of them has its roots in the Sharkovski0 Theorem. This beautiful theorem describes the possible sets of periods of all cycles of maps of an interval into itself. Another direction has its main objective in measuring the complexity of a system, or the amount of chaos present in it. A good way of doing this is to compute topological entropy of the system. The aim of this book is to provide graduate students and researchers with a unified and detailed exposition of these developments for interval and circle maps. Many comments are added referring to related problems, and historical remarks are made.
Reviews / Votes
"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." A Quas Mathematics AbstractsMore details
Series
Language
English
Place of publication
Singapore
Singapore
Target group
College/higher education
Professional and scholarly
ISBN-13
978-981-02-1344-2 (9789810213442)
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Schweitzer Classification
Persons
Content
Part 1 Preliminaries: general notation; graphs, loops and cycles. Part 2 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. Part 3 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. Part 4 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.