KöMaL 2018 A735

For any function $f\colon [0,1]\to[0,1]$, let $P_n(f)$ denote the number of fixed points of the function $\underbrace{f\big(\ldots f}_n(x)\ldots\big),$ i.e., the number of points $x\in[0,1]$ satisfying $\underbrace{f\big(\ldots f}_{n}(x)\ldots\big)=x$. Construct a piecewise linear, continuous, surjective function $f\colon [0,1]\to[0, 1]$ such that for a suitable number ${2<A<3}$, the sequence $\frac{P_n(f)}{A^n}$ converges.