Bernoulli Map properties I am referring to the function stated here http://en.wikipedia.org/wiki/Dyadic_transformation
This map is defined on $[0,1]$ by $f_n(x)=nx [mod 1]$
There are three things I do not quite understand, may you can help me with that.Lets strt with the well known case $n=2$, i.e $f_2(x)=2x [mod 1]$, this means when we consider the fractional part of some value in each step we move the coma one step to the right. An example for a 2-cycle would be $\{1/3,2/3\}$
My first question: How can I find all 3-cycles of $f_2$ without simply guessing the numbers?
My second question: What does $f_3$ means? Is this simply the representation from numbers with base 3?
My third question (I have no idea about that): Do you have an idea for a non periodic orbit of the Bernoulli map that is dense in $[0,1)$?
 A: Suppose that $x = \sum_{k = 1}^{\infty} d_k / 2^k$ is the dyadic expansion ($d_k \in \{0,1\}$) of the point $x \in [0,1)$, and assume as well that the dyadic expansion is unique.
Then, $f_2(x)$ will have the (unique; one must check this!) dyadic expansion
$$
\sum_{k = 1}^{\infty} \frac{d_{k + 1}}{2^k}
$$
Put differently, $f_2$ acts as the leftward shift operator on the space of sequences $(d_k)_k$ of 0's and 1's.
1st Question: What is a 3-cycle? Well, if $x$ has a unique dyadic expansion and happens to be a 3-cycle under $f_2$, then what can you say about its expansion $d_1, d_2, \cdots$? Since $f_2^3 x = x$, you must have that $d_{k + 3} = d_k$!
2nd Question: $f_3$ is going to be the same kind of shift, except that it refers to shifting the base 3 expansion instead of the base 2 expansion.
3rd Question: There is certainly such an orbit! The idea is to construct $x$ so that for any fixed initial sequence $d_1', \cdots, d_q'$ of any length and any $N$ large, there is an $n > N$ such that the dyadic expansion of $f_2^n x$ agrees with $d_1', \cdots, d_q'$ out to the $q$-th place. Can you see why such a point should have a dense orbit? How would you construct such an $x$?
Another thing: What about the nonunique expansions? When does this happen, and what do the dynamics of $f_2$ look like on them? Hint: think about $.9999 = 1.0$ in base 10; what's the analogue in base 2?
