To prove - cover of union of backwards orbits of tent map is I

Question

I read, that

$$ \overline{\bigcup^\infty_{n=0} T^{-n}(0)} = I, $$

(where $T$ is Tent map and $I = [0; 1]$) but I don't understand why.

Could you, please, show me the proof of this?

Answer 1

We have $T^{-1}(\{0\}) = \{0,1\}$. To compute the higher inverse images, note, that $T^{-1}(\{x\}) = \{\frac x2,1-\frac x2\}$ for $x\in[0,1]$. Therefore $$ T^{-2}(\{0\}) = \left\{0,\frac 12, 1\right\} $$ By induction, we see that $$ T^{-n}(\{0\}) = \left\{ \frac{k}{2^{n-1}} : k = 0,\ldots, 2^{n-1}\right\}$$ This gives us that $\bigcup_n T^{-n}(\{0\})$ consists of all dyadic fractions, which form a dense subset of $[0,1]$.