Summary
We give a general construction of iteration groups of continuous functionsG:= {f t,t ∈ ℝ}} on an interval (a, b) such thatf t (x) ≠ x forx ∈ (a, b) providedf t ≠ id. Two cases may occur
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(i)
For everys, t ∈ ℝ such thatf s ≠ id andf t ≠ id there existn, m ∈ ℤ such that|n| + |m| ≠ 0 andf ns =f mt
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(ii)
There exists, t ∈ ℝ such thatf s ≠ id andf t ≠ id and for everyn, m ∈ ℤ, |n| + |m| ≠ 0, f ns ≠ f mt.
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References
Kuczma, M.,Functional equations in a single variable. PWN-Polish Scientific Publishers, Warszawa, 1968.
Kuratowski, K. andMostowski, A.,Teoria mnogości (Set theory). PWN-Polish Scientific Publishers, Warszawa, 1966.
Tabor, J.,Rational iteration groups. Wyż. Szkola Ped. Krakow. Rocznik Nauk.-Dydakt. Prace Mat.11 (1985), 153–176.
Targoński, Gy.,New directions and open problems in iteration theory. [Ber. Math.-Stat. Sekt. Forchungszentrum Graz, Nr. 229]. Research Inst., Graz, 1984.
Vincze, E.,Über die Charakterisierung der assoziativen Funktionen von mehreren Veränderlichen. Publ. Math. Debrecen6 (1959), 241–253.
Weitkamper, J.,Embeddings in iteration groups and semigroups with nontrivial units. Stochastica8 (1983), no. 3, 175–195.
Zdun, M. C.,Note on commutable functions. Aequationes Math.36 (1988), 153–164.
Zdun, M. C.,On simultaneous Abel equations. Aequationes Math.38 (1989), 163–177.
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Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.
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Zdun, M.C. The structure of iteration groups of continuous functions. Aeq. Math. 46, 19–37 (1993). https://doi.org/10.1007/BF01833995
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DOI: https://doi.org/10.1007/BF01833995