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

Misiurewicz, Michał; Nitecki, Zbigniew

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.

Reviewer: S.Yu.Pilyugin (St.Petersburg)

MSC:

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

Keywords:

interval map; combinatorial pattern