Means of iterates

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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 kn is odd, then such functions consist of at most three affine pieces.

Keywords

Continuous solution Iterate Polynomial-like iterative equation 

Mathematics Subject Classification

Primary 39B22 Secondary 26A18 39B12 

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Diebold Nixdorf BPOKatowicePoland
  2. 2.Instytut MatematykiUniwersytet Śla̧skiKatowicePoland

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