Means of iterates

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 k 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.2010 Mathematics Subject Classification. Primary: 39B22. Secondary: 26A18, 39B12.