Periodic solutions of the Poincare functional equation: Existence

Let I be a compact real interval, k > 1 an integer and f : I -> I a continuous surjection such that: (i) f achieves its absolute extrema at exactly k + 1 points, and on the k remaining intervals f is strictly monotone, (ii) the set of all points among which the iterations of f attain their absolute extrema is dense in I. We show that the above conditions on f are necessary and sufficient for the Poincare functional equation psi(kx) = f(psi(x)) to have a cosine-like solution psi : R -> R. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The conditions mentioned above are necessary and sufficient for the Poincare functional equation to have a cosine-like solution.