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 2^{n} 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 2^{n} polynomials of degree *n* in $\mathbb{Z}_2[x]$. 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

nin $\mathbb{Z} _ 2[x]$ = $\frac{1}{n}\sum_{d|n} \mu(d)2^{n/d}.$

**Question**: it's obvious by now: is there a natural and structured bijection between periodic orbits of *f* and irreducible polinomials in $\mathbb{Z}_2[x]$? How is interpreted the structure of one context when transported ni the other?

(**rmk**: of course, analogous identities hold for any p > 2)