MathDB

3

Part of 1998 IMC

Problems(2)

f(x)=2x(1-x)

Source: IMC 1998 day 1 problem 3

11/1/2005
Let f(x)=2x(1−x),x∈Rf(x)=2x(1-x), x\in\mathbb{R} and denote fn=f∘f∘...∘ff_n=f\circ f\circ ... \circ f, nn times. (a) Find lim⁡n→∞∫01fn(x)dx\lim_{n\rightarrow\infty} \int^1_0 f_n(x)dx. (b) Now compute ∫01fn(x)dx\int^1_0 f_n(x)dx.
limitintegrationreal analysisreal analysis unsolved
IMC 1998 Problem 9

Source: IMC 1998 Day 2 Problem 3

10/28/2020
Given 0<c<1 0< c< 1, we define f(x)={xcx∈[0,c]1−x1−cx∈[c,1]f(x) = \begin{cases} \frac{x}{c} & x \in [0,c] \\ \frac{1-x}{1-c} & x \in [c, 1] \end{cases} Let fn(x)=f(f(...f(x)))f^{n}(x)=f(f(...f(x))) . Show that for each positive integer nn, fnf^{n} has a non-zero finite nunber of fixed points which aren't fixed points of fkf^k for k<nk< n.
Fixed pointreal analysis