Abstract
We determine continuous bijections f, acting on a real interval into itself, whose k-fold iterate is the quasi-arithmetic mean of all its subsequent iterates from \(f^0\) up to \(f^n\) (where \(0\leqslant k\leqslant n\)). Namely, we prove that if at most one of the numbers k, n is odd, then such functions consist of at most three affine pieces.
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Dedicated to Professor Karol Baron on his 70th birthday.
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Draga, S., Morawiec, J. Means of iterates. Aequat. Math. 93, 21–35 (2019). https://doi.org/10.1007/s00010-018-0550-y
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DOI: https://doi.org/10.1007/s00010-018-0550-y