I want to find all the periodic points of the following diffeomorphism of the circle:
$f(x) = x + \frac{1}{4} + \frac{1}{10} \sin(8 \pi x) \mod 1$
Where a periodic point is $p$ such that $f^n(p) = p$ for some $n \in \mathbb{N}$. I have shown the rotation number $\rho (f) = \frac{1}{4}$ (I think) so I know that periodic points exist (as the rotation number is rational) but that none are fixed points, i.e. no such $p$ for $n = 1$ (as rotation number is not zero).
I tried working from the definition but I get a horrible nest of sine functions. In general finding periodic points of functions like this doesn't appear to be well documented in my course notes, the recommended book or indeed the internet (from what I can find) so any help at all would be appreciated. I'm looking for a general method, not just the solution for this specific diffeomorphism, but really any help is great.
This isn't homework if anyone was wondering, it's revision and I'm more interested in the mathematics behind the general method for a solution than getting the specific solution of this question. Hints definitely welcome.
EDIT: Forgot to add, as for a lift $F$ of $f$ we have $deg(f) = F(x + 1) - F(x) = 1$, we know the diffeomorphism is orientation preserving.
Since you know what is the rotation number, there is simple way. Recall that the rotation number is $p/q$ with $p$ and $q$ coprime if and only if $f$ has a periodic point of period $q$. So you should concentrate on finding a point of period $4$, and a simple inspection shows $0$ has period $4$.
Other than that, the only question remaining is whether there are other orbits with period $4$ (for sure there are no orbits with other periods!). Indeed there are three other such orbits, that of $1/8$ and two others.