Means of iterates
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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.
KeywordsContinuous solution Iterate Polynomial-like iterative equation
Mathematics Subject ClassificationPrimary 39B22 Secondary 26A18 39B12
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