SCHWEITZER 2018 Problem 8

Does there exist a piecewise linear, continuous, surjective mapping $f: [0,1]\to [0,1]$ such that $f(0)=f(1)=0$ , and for all positive integer $n$ , $$2.0001^{(n-10)} <P_n(f)<2.9999^{(n+10)}$$ holds, where $P_n(f)$ is the number of points $x$ such that $\underbrace{f(\dotsc f}_n(x)\dotsc )=x$ ?