Problem A. 735. (November 2018)

A. 735. For any function \(\displaystyle f\colon [0,1]\to[0,1]\), let \(\displaystyle P_n(f)\) denote the number of fixed points of the function

\(\displaystyle \underbrace{f\big(\ldots f}_n(x)\ldots\big), \)

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

Based on the 8th problem of the Miklós Schweitzer competition, 2018

(7 pont)

Deadline expired on December 10, 2018.

Statistics:

4 students sent a solution.
7 points:	Schrettner Jakab.
2 points:	3 students.

Problems in Mathematics of KöMaL, November 2018