So in my Dynamical System course there is two problems about $2^n$ that I don't know how to solve first showing that there exists n such that $2^n$ written in 10 decimal forms starts with $123456789$ and second computing the density of n such that 2^n starts with 5. I can't seem to find the connection between $2^n$ and what we do in Dynamical systems (Basic topological definitions and theorems and Basic Ergodicity). Maybe we use the space $\{0,1\}^\mathbb{N}$ but I dont know how. I would appreciate any help.
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1Think about iterating the map $x\mapsto\begin{cases}2x & x< 1/2 \ \frac{2x}{10} & x \ge 1/2 \end{cases}$. – hmakholm left over Monica Dec 18 '13 at 18:46
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Thanks for the hint so the first problem is solved if we can show that the orbit of 1 is dense but I dont see why the orbit is dense – omar Dec 18 '13 at 19:35
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1x @omar: It may be easier to rephrase the iteration in terms of the logarithms of the $x$s instead. Then remember that $\log_{10} 2$ is irrational. – hmakholm left over Monica Dec 18 '13 at 19:41
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I dont see what you mean by rephrasing the iteration in terms of $log$. – omar Dec 18 '13 at 19:58
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x @omar: Define $y_n = \log x_n$, and rewrite the iteration formula such that it expressed $y_n$ as a function of $y_{n-1}$. – hmakholm left over Monica Dec 18 '13 at 20:15
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Ok it is clear now thanks.(Do you have any suggestion for the second question) – omar Dec 18 '13 at 20:43
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You can reuse much of the work from the first. Hint: the answer is $\log_{10} 6 - \log_{10} 5 $. – hmakholm left over Monica Dec 18 '13 at 20:47