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On eventual coloring numbers. (English) Zbl 1331.54043

Let \(f: X \to X\) be a fixed-point free map of a separable metric space \(X\). A subset \(C\subset X\) is eventually colored within \(p\) of \(f\) if \(\bigcap_{i=0}^p f^{-i}(C)=\emptyset\). A cover \( \mathcal C\) of \(X\) is called an eventual coloring of \(f\) within \(p\) if each element \(C \in \mathcal C\) is eventually colored within \(p\) of \(f\). The minimal cardinality \(C(f, p)\) of all closed (or open) eventual colorings of \(f\) within \(p\) is called the eventual coloring number of \(f\) within \(p\). The coloring number of \(f\) is \(C(f,1)\).
The authors build on their previous work by providing an index \(\psi_n(n)\), which they use to bound the eventual coloring number. The main result of the paper states the following: if \(\dim X=n < \infty\) and the set of periodic points of \(f\) is at most zero-dimensional then \[ C(f, \psi_n(k))\leq n + 3 - k \] for each \(k=0, 1, \dots, n+1\).
Reviewer: Ziga Virk (Litija)

MSC:

54H20 Topological dynamics (MSC2010)
54F45 Dimension theory in general topology
55M10 Dimension theory in algebraic topology
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
54C05 Continuous maps
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