MathDB

Original in Hungarian; translated with Google translate; polished by myself.
Let

f: [0, \infty) \to [0, \infty)

be a function that is linear between adjacent integers, and for

n = 0, 1, \dots

satisfies

f(n) = \begin{cases} 0, & \textrm{if }2\mid n,\\4^l + 1, & \textrm{if }2 \nmid n, 4^{l - 1} \leq n < 4^l(l = 1, 2, \dots).\end{cases}

Let

f^1(x) = f(x)

, and

f^k(x) = f(f^{k - 1}(x))

for all integers

k \geq 2

. Determine the values of

\liminf\nolimits_{k\to\infty}f^k(x)

and

\limsup\nolimits_{k\to\infty}f^k(x)

for almost all

x \in [0, \infty)

under Lebesgue measure.
(Not sure whether the last sentence translates correctly; the original: Határozzuk meg Lebesgue majdnem minden

x\in [0, \infty)

-re a

\liminf\nolimits_{k\to\infty}f^k(x)

és

\limsup\nolimits_{k\to\infty}f^k(x)

értékét.)

Problem Statement