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Does there exist a function

Source: Miklós Schweitzer 2018 P8

November 10, 2018
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Problem Statement

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