What is the difference between a period $n$ point, and a point of least period $n$?
Simply what is the definition of the two of them, and how do they differ. I think I have a rough idea of one but not sure how it differs.
My thinking, for $n=2$ say...
If $f^2(x) = x$ then we have a period $2$ point. A point of least period $2$ I'm guessing, simply is an orbit where, for example, $(x_1,x_2,x_3)$ are the points in the orbit, $f^2(x_1) = x_1$ but $f^3(x_2) = x_2$ and $f^3(x_3) = x_3$ (Maybe this example does not work, but hopefully you get the idea?)
Either way what is a good definition of both for general n? And some trivial examples?
Thank you in advance!
A point with period $n$ is also a point of period $kn$ for any natural number $k$. In particular, a point with period $n$ is also a point of period $2n$, but is certainly not a point of least period $2n$, since $n < 2n$.
Since you ask for the definition:
A point of period $n$ is a point such that $f^{n}(x) = x$. A point of least period $n$ is a point such that $f^{n}(x) = x$ but $f^{k}(x) \neq x$ for any natural number $k < n$.