Combinatorial patterns for maps of the interval. (English) Zbl 0745.58019

Mem. Am. Math. Soc. 456, 112 p. (1991).
Let \(f\) be a continuous map of a closed interval \(I\) to itself. The authors define a combinatorial pattern \(\theta\) of \(f\). They say that a combinatorial pattern \(\theta\) forces another combinatorial pattern \(\eta\) if every continuous map \(f: I\to I\) which exhibits \(\theta\) also exhibits \(\eta\).
Some criteria are given for deciding if \(\theta\) forces \(\eta\) in any specific case. Extensions and reductions of patterns are studied. The authors investigate a weakening of the notion of extension, which they call combinatorial shadowing. The relation between entropy estimates arising from different patterns is explored.


37B40 Topological entropy
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
54H20 Topological dynamics (MSC2010)
26A18 Iteration of real functions in one variable
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