VTRMC 2018 Problem 7

A continuous function $f : [a,b] \to [a,b]$ is called piecewise monotone if $[a, b]$ can be subdivided into finitely many subintervals $$I_1 = [c_0,c_1], I_2 = [c_1,c_2], \dots , I_\ell = [ c_{\ell - 1},c_\ell ]$$ such that $f$ restricted to each interval $I_j$ is strictly monotone, either increasing or decreasing. Here we are assuming that $a = c_0 < c_1 < \cdots < c_{\ell - 1} < c_\ell = b$ . We are also assuming that each $I_j$ is a maximal interval on which $f$ is strictly monotone. Such a maximal interval is called a lap of the function $f$ , and the number $\ell = \ell (f)$ of distinct laps is called the lap number of $f$ . If $f : [a,b] \to [a,b]$ is a continuous piecewise-monotone function, show that the sequence $( \sqrt[n]{\ell (f^n )})$ converges; here $f^n$ means $f$ composed with itself $n$ -times, so $f^2 (x) = f(f(x))$ etc.