Please give some counterexamples, some definition could be found here. Eventually periodic point and homeomorphism. An eventually periodic point must be an asymptotically periodic point?
1.a recurrent point meanwhile not a transitive point
2.a recurrent point meanwhile not a periodic point
3.a recurrent point meanwhile not a eventually periodic point
I think that a recurrent point is abstract noun, so I need some examples to visualize it in my mind.
I suspect that you're talking about topological dynamics, where you have a continuous function $T$ from a space $X$ into itself, and using the following definitions:
A "recurrent point" is a point $p$ such that for every neighbourhood $U$ of $p$, there are infinitely many $n$ such that $T^n(p) \in U$.
A "transitive point" is a point $p$ such that for every nonempty open set $A$ there are infinitely many $n$ such that $T^n(p) \in A$.
A "periodic point" is a point $p$ such that for some positive integer $n$, $T^n(p) = p$.
An "eventually periodic point" is a point $p$ such that there exist positive integers $m < n$ such that $T^m(p) = T^n(p)$.
An example for all your questions is provided by an irrational rotation of a disk. Every point is recurrent, but no points are transitive, and no point except the origin is periodic or eventually periodic.
