Two kinds of analytical solutions to Feigenbaum–Kadanoff–Shenker equation (Q6581929)

The FKS equation \N\[\N\begin{cases} \Nf(f(\alpha^2x))=\alpha f(x),\\\Nf(0)=1, \N\end{cases}\N\]\Nwhere \(\alpha \in (-1,0)\) and \(f:[-1,1] \rightarrow [-1,1]\) is an even unimodal function was introduced by \textit{M. J. Feigenbaum} et al. [Phys. D. 5, No. 2--3, 370--386 (1982; \url{doi:10.1016/0167-2789(82)90030-6})]. Although multiple authors have made progress toward solutions of the FKS equation with numerical simulation techniques, an exact solution was obtained by \textit{Y.-G. Shi} [Aequationes Math. 93, No. 5, 919--925 (2019; Zbl 1473.39039)] using a method suggested by \textit{L. Yang} and \textit{J. Zhang} [Sci. Sin., Ser. A 29, 1252--1262 (1986; Zbl 0641.58017)]. \N\NThe paper reviewed here uses a different method to construct continuous piecewise linear solutions and continuous piecewise fractional linear solutions to the FKS equation. Examples are provided.