For the map $T: [0,1]\to [0, 1]$ defined by $Tx=10x\pmod{1}$,
how to use the decimal expansion to construct a $x$, such that the orbit of $x$, say $\theta_x=\{T^nx: n\geq 0\}$ is dense in $[0, 1]$ and $T^px\to 0$ as $p\to \infty$, where $p$ runs over all prime number ?
On
If I'm not mistaken you would have to prove first (or assume) that there are arbitrarily long gaps between prime numbers if you really want $T^p x \rightarrow 0$ and not just the $liminf$. In these gaps you can insert all possible patterns of expansions (to get density) while you insert sequences of zeros elsewhere. You will also need to make sure you have infinitely long sequences of zeros but this is easy to achieve (you can just not use all the gaps between primes).
I guess I'm not so clear above, but along with Wong's answer I hope it helps...
Some hints: You can think of $T$ as a shift map, where you shifting up the digits of a decimal expansion. The set of all numbers with finite decimal expansions (i.e. having all zeros after a certain point) is countable and dense in $[0,1]$, so you can "hide" them inside the decimal expansion of an appropriate $x$, to be revealed by repeated application of $T$. Furthermore, to make $T^p x \rightarrow 0$, you'll want to "hide" longer and longer strings of zeros inside the decimal expansion.