The Feigenbaum functional equation and periodic points
The behavior of solutions of the Feigenbaum functional equation \[ (1)\quad f(f(\lambda x))+\lambda f(x)=0,\quad \lambda =f(-1) \] is studied. The following result is, particularly, proved. Theorem. If f is an even unimodal solution of (1) with \(-1<f(1)<0\), then f has a unique fixed point, say \(x_ 1\), where \(x_ 1>0\), \(\lambda <x_ 1<f(\lambda)\). If f does not have a period 2 point in the interval (\(\lambda\),f(\(\lambda)\)), then \(x_ 1\) is an unstable fixed point of f. Furthermore in this case, if \(x\in (\lambda,f(\lambda))\), \(x\neq x_ 1\), then the orbit of x leaves this interval in finitely many steps, and there is no periodic point of any period in (\(\lambda\),f(\(\lambda)\)).
scientific article; zbMATH DE number 3953445