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Suppose $f:[0,1]\rightarrow\mathbb{R}$ is defined by $$ f(x) = \left\{ \begin{array}{rl} 3x &\mbox{ if $0\leq x<\frac{1}{2}$} \\ 3x-\frac{3}{2} &\mbox{if $\frac{1}{2}\leq x\leq 1$} \end{array} \right. $$

Suppose $n\in \mathbb{N}$ and $y\in f^{n}([0,1])$ where $f^{n}:f^{n-1}([0,1])\rightarrow\mathbb{R}$ is composition $n$ times. Let $g_{n}=f^{n}_{|[0,1]}$ ($g_{n}$ is $f^{n}$ restricted to $[0,1]$). Is it possible to estimate the value $|g^{-n}(\{y\})|$ by a formula depending on $n$ and $y$?

Note: $|\ |$ is cardinality and $g^{-n}(\{y\})$ is the set of pre images of $y$.

I cant find a pattern in this problem, please i need help.


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What does $f^n$ mean? $f$'s range is larger than its domain, so how can it be iterated? – fgp Oct 12 '12 at 19:51
You are right, let me try correct it. – Tomás Oct 12 '12 at 19:59

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