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Combinatorial Patterns for Maps of the Interval
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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 them...
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.
Table of contents:
Adjusted maps; Equivalent and essential patterns; Reductions and extensions of patterns; Irreducible patterns and Markov graphs; Horseshoe patterns and fold type; Extensions of cycles; Combinatorial shadowing; Transitive patterns; Representatives of cycles; Forcing and degree; Entropy estimates.
Table of contents:
Adjusted maps; Equivalent and essential patterns; Reductions and extensions of patterns; Irreducible patterns and Markov graphs; Horseshoe patterns and fold type; Extensions of cycles; Combinatorial shadowing; Transitive patterns; Representatives of cycles; Forcing and degree; Entropy estimates.