ISSN:
1420-8903
Keywords:
Primary 26A18
;
Secondary 39B05, 06A10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor: “Letf: E → E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation:ϕ ≦ ψ if and only ifϕ is an iterative root ofψ. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is ‘yes’. But in general the question is open.” Here we prove that there exists a three-element decomposition {Φ i ;i = 1, 2, 3} of the setE E with blocks Φi of the same cardinality 2cardE such that the functions from Ф1 do not possess any proper iterative root, the quasi-ordering ≦ is not antisymmetric onF(f) for anyf ∈ Φ2, and ≦ is an ordering onF(f) for anyf ∈ Ф3. Iff is a strictly increasing continuous self-bijection ofE, then the relation ≦ is an ordering onF(f) ifff is different from the identity mapping of the setE.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01833938
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