Which functions have a list for all periodic points of them?

Question

I was searching for a method that finds all periodic points of a given function e.g. $f(x)=x-x^2$ on its domain. As @Did explained in comment it's too hard even though for polynomials of degree 2.
So I change the question to list and categorize functions with all their periodic points.
Definition: The point $x$ is a periodic point of period $n$ if $f^n(x) = x$. (I will add functions which I know them as soon as I can.) Please add new functions if you know.

Answer 1

This is a huge question that deserves a long answer:

Part 1: Some functions whose periodic points are known

There are some standard easy functions whose periodic points you can just write down; these examples tend to show up as "exceptions" to a lot of theorems in complex dynamics.

Example 1: Power maps $f(x) = x^d$, where $d\in \mathbb{Z}$. Of course the most interesting case is when $d\geq 2$. Then you compute $f^{n}(x) = x^{d^n}$, so the $n$-periodic points are those which satisfy the equation $x^{d^n}=x$. Thus the $n$-periodic points are $0$ and the $(d^{n}-1)$st roots of unity. If you are looking at $f$ as a dynamical system on the Riemann sphere $\mathbb{C}\cup\{\infty\}$, then $\infty$ is also a fixed point.

Sometimes we care about not just periodic points but preperiodic points as well (these are points whose orbits are finite). For the case of $f(x) = x^d$, $d\geq 2$, the preperiodic points are all roots of unity, as well as $0$ and $\infty$.

Example 2: Chebyshev polynomials. Let $d\geq 0$ be an integer. Then you can show there is a unique polynomial $T_d$ which satisfies $$T_d(x + x^{-1}) = x^d + x^{-d}$$ This polynomial is called the $d$th Chebyshev polynomial. You can show that $T_0 = 2$, $T_1 = x$, and that $T_d = xT_{d-1} - T_{d-2}$ for each $d\geq 2$. The reason you can compute the periodic points of these pretty easily is because of their close relationship with the power maps in example 1. Specifically, if you let $f_d$ be the power map $f_d(x) = x^d$ and let $\phi(x) = x + x^{-1}$, then the defining relation (displayed above) for the Chebyshev polynomials can be restated as $\phi\circ f_d = T_d\circ \phi$, that is to say, $\phi$ is a semiconjugacy between $f_d$ and $T_d$. It's not a conjugacy, because $\phi$ is a degree $2$ map, so you have to do some work still, but it's possible to show that the periodic points of $T_d$ are precisely the points of the form $\phi(\zeta)$, where $\zeta$ is a periodic point of $f_d$. A similar statement holds for preperiodic points. Note $\phi(\zeta)$ is $2\Re(\zeta)$ for any root of unity $\zeta$.

Example 3: Lattès Examples. I won't say much about these, but the idea is similar to the Chebyshev polynomial construction. The Chebyshev polynomials were obtained from the power maps by a semiconjugacy; Lattès examples are obtained from certain endomorphisms of elliptic curves by a semiconjugacy. The fact that the periodic points of the endomorphisms of the elliptic curves are easy to compute makes the periodic points of Lattès examples easy to compute as well. I should say, though, Lattès examples are rational maps, not polynomials.

Part 2: Understanding the distribution of periodic points

Suppose you just wanted to have a rougher idea of where in the complex plane the periodic points of a polynomial $f(x)$ are located. One first result along these lines is the following.

Theorem: Suppose $f$ has degree $d\geq 2$. Then all but at most $2d-2$ periodic points of lie in the Julia set of $f$, and in fact these are dense in the Julia set of $f$.

So to get an idea where the periodic points of $f$ are located, you can have a computer draw the Julia set of $f$, and that will give you a pretty good picture of where all but a few periodic points are located. The good news about this is that people have come up with pretty fast ways (using potential theory) to have a computer draw the Julia set of a polynomial. There are more precise versions of this as well, like the following.

Theorem: Suppose $f$ has degree $d\geq 2$. Then there is a canonical probability measure $\mu$ supported on the Julia set which gives the asymptotic distribution of the $n$-periodic points of $f$ as $n\to \infty$.

Part 3: Using arithmetic to locate preperiodic points

More recently, people have started studying problems like this using the tools of number theory. A full answer here would be too long, so I'll just give a rough idea of what people do in a very special case.

Assume that $f(x)$ is a polynomial of degree $d\geq 2$ with rational coefficients, and that we're only interested in finding rational periodic points (everywhere here it is possible to replace rational algebraic, but I'm going to stick to the rational case for simplicity). It is possible to define in a natural way a function $h_f\colon \mathbb{Q}\to [0, \infty)$, which is typically called the canonical height function of $f$, which has the remarkable property that for a rational number $r$, one has $h_f(r) = 0$ if and only if $r$ is preperiodic for $f$. So to find preperiodic points, we just need to find those rational numbers $r$ for which $h_f(r) = 0$.

The catch here is that $h_f$ is easy to define but hard to compute. But it has another remarkable property: there is another, simpler function $h\colon \mathbb{Q}\to [0, \infty)$ which has nothing to do with $f$, called the naive height function, for which $h_f - h$ is bounded. For a rational number $r = a/b$ written out in lowest terms, this naive height is simply $h(r) := \log\max\{|a|, |b|\}$.

So, assume that you were able to cook up a constant $C>0$ for which $|h_f - h| \leq C$. Then any rational preperiodic point $r$ of $f$ would satisfy $h_f(r) = 0$, so that $h(r) = \log\max\{|a|, |b|\} \leq C$. This gives you bounds on the possible numerators and denominators of $r$. Ok, in fairness, it's a terrible bound: $|a|, |b| \leq e^C$ grows quite rapidly with $C$. But still, it's something!

Reference: The best reference I can think of that discusses all of the things I mention here is the book The Arithmetic of Dynamical Systems by Silverman.