MathDB

Does there exist a piecewise linear continuous function

f:\mathbb{R}\to \mathbb{R}

such that for any two-way infinite sequence

a_n\in[0,1]

n\in\mathbb{Z}

, there exists an

x\in\mathbb{R}

with

\limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n

for all

n\in\mathbb{Z}

, where

f^k=f\circ f\circ \dots\circ f

stands for the

k

-fold iterate of

f

Problem Statement