Counterexamples of dynamical systerms

Question

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.

Answer 1

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:

1. A "recurrent point" is a point $p$ such that for every neighbourhood $U$ of $p$, there are infinitely many $n$ such that $T^n(p) \in U$.

2. A "transitive point" is a point $p$ such that for every nonempty open set $A$ there are infinitely many $n$ such that $T^n(p) \in A$.

3. A "periodic point" is a point $p$ such that for some positive integer $n$, $T^n(p) = p$.

4. 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.