Combinatorial Patterns for Maps of the Interval (Memoirs of the AMS) (Memoirs of the American Mathematical Society) - Softcover

Misiurewicz, Michal; Nitecki, Zbigniew

Softcover

ISBN 10: 0821825135 ISBN 13: 9780821825136

Publisher: American Mathematical Society, 1992

Synopsis

In recent years, motivated by Shrkovskii's theorem, researchers have realized that a good deal of information about the dynamics of a map on the interval can be deduced from the combinatorial structure of its periodic orbits. This data can be formulated as a "forcing" relation between cyclic permutations (representing "orbit types" of periodic orbits). The present study investigates a number of new features of this relation and its generalization to multicyclic permutations
(modelling finite unions of periodic orbits) and combinatorial patterns (modelling finite invariant sets). A central theme is the role of reductions and extensions of permutations. Results include: (i) a "combinatorial shadowing theorem" and its application to approximating permutations by cycles in the forcing relation; (ii) the distribution of different representatives of a given cycle in one (adjusted) map; (iii) characterization of the forcing-maximal permutations and patterns of fixed
degree;
and (iv) a calculation of the asymptotic growth rate of the maximum entropy forced by a permutation of given degree.

Edition details

• Publisher: American Mathematical Society
• Publication date: 1992
• Language: English
• ISBN 10: 0821825135
• ISBN 13: 9780821825136
• Binding: Paperback