TITLE: Periodic orbits and polynomials
QUESTION [16 upvotes]: There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" f:[0,1]→[0,1] with parameter 2, that is 

f(x):=2min(x,1-x).

Clearly, it has 2 fixed points, and more generally, for any positive integer n, there are 2n periodic points of period n (it's easy to count them as they are fixed points of the n-fold iteration of f, which is a piecewise linear function oscillating up and down between 0 and 1 the proper number of times). To count the number of periodic orbits of minimal period n, a plain and standard application of the Moebius inversion formula gives

Number of n-orbits of (I,f) =
  $\frac{1}{n}\sum_{d|n} \mu(d)2^{n/d}.$

(rmk: any function with similar behaviour would give the same result, e.g. f(x)=4x(1-x),...&c.)
Now let's leave for a moment dynamical systems and consider the following enumeration in the theory of finite fields.
fact 2 Clearly, there are 2n polynomials of degree n in $\mathbb{F}_2[x]$. With a bit of field algebra it is not hard to compute the number I(n) of the irreducible ones. One can even make a completely combinatorial computation, just exploiting the unique factorization, expressed in the form:
$\frac{1}{1-2x}=\prod_{n=1}^\infty (1-x^n)^{-I(n)}.$ 
One finds:

Number of irreducible polynomials of
  degree n in $\mathbb{F} _ 2[x]$ =
  $\frac{1}{n}\sum_{d|n} \mu(d)2^{n/d}.$

Question: it's obvious by now: is there a natural and structured bijection between periodic orbits of f and irreducible polinomials in $\mathbb{F}_2[x]$? How is interpreted the structure of one context when transported ni the other?
(rmk: of course, analogous identities hold for any p > 2)

REPLY [11 votes]: How I see it is every fixed point comes from an equation $T^nx=x$. Denoting $f(x)=2x$ and $g(x)=2(1-x)$ we see that this corresponds to solving equations 
$$h_1\circ h_2\circ\cdots \circ h_n (x)=x$$ where $h_i\in \{f,g\}$. This is geometrically the intersection of two lines and thus gives a unique solution, and therefore a correspondence between periodic points of period $n$ and binary strings of length $n$. Now since concatenating a binary string $L$ with itself clearly gives the same $x$ with $x=L(x)=L\circ L(x)$ and you have a shift operator $L\circ h(x)=x \implies h\circ L(y)=y$ where $y=h(x)$ you get a bijection between periodic orbits and aperiodic cyclic sequences of zeros and ones, which are well known to be in bijection with the irreducible polynomials of $\mathbb F_2 [x]$.