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Convergence of a Piecewise-Monotone Function Sequence

Source: 2018 VTRMC P7

January 8, 2023
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Problem Statement

A continuous function f:[a,b]→[a,b]f : [a,b] \to [a,b] is called piecewise monotone if [a,b][a, b] can be subdivided into finitely many subintervals I1=[c0,c1],I2=[c1,c2],…,Iℓ=[cℓ−1,cℓ]I_1 = [c_0,c_1], I_2 = [c_1,c_2], \dots , I_\ell = [ c_{\ell - 1},c_\ell ] such that ff restricted to each interval IjI_j is strictly monotone, either increasing or decreasing. Here we are assuming that a=c0<c1<⋯<cℓ−1<cℓ=ba = c_0 < c_1 < \cdots < c_{\ell - 1} < c_\ell = b. We are also assuming that each IjI_j is a maximal interval on which ff is strictly monotone. Such a maximal interval is called a lap of the function ff, and the number ℓ=ℓ(f)\ell = \ell (f) of distinct laps is called the lap number of ff. If f:[a,b]→[a,b]f : [a,b] \to [a,b] is a continuous piecewise-monotone function, show that the sequence (ℓ(fn)n)( \sqrt[n]{\ell (f^n )}) converges; here fnf^n means ff composed with itself nn-times, so f2(x)=f(f(x))f^2 (x) = f(f(x)) etc.
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