Let $X$ be a path connected topological space and $f:X\rightarrow X$ be a continiuous embedding (i.e. $f$ is homeomorphic onto its image). A point $x\in X$ is a periodic point of $f$ with period $n$ when $n$ is the smallest positive integer s.t. $f^n(x)=x$
Now I've had an idea and I couldn't find any counter examples, but it seems too good to be true. I thought that maybe the periodic points of such a map can only possibly have finitely many different periods. That is there is a positive integer $m$ s.t. for all $n>m$ there are no periodic points of $f$ with period $n$. My (first) question is if this is true.
If this really happens to be true I've thought of an even stronger statement that I could not find any counter examples to either, though I really can't imagine that it could be true. The statement is that every such map can only have periodic points of two different periods and maybe even that one of those periods has to be $1$ if there really are two different periods. My second question is then wether this is true as well.
If these statements aren't true, are there conditions on the space or the map that make them true? For example, are they true for automorphisms of the space $X$?
Even if these statements are true, I have absolutely no idea how I could prove them. (I did manage to prove that the periodic points of such maps from $\mathbb{R}$ to $\mathbb{R}$ can only have a period of $1$ or $2$, but the proof made heavy use of the fact that the real numbers are totally ordered). Are there any methods to prove statement similar to these?
I'm sorry for the many questions. I would already be very glad if somebody can answer any one of them.