Re: Non-wandering and recurrent points in a dynamical system

In article
<507dddcf-14e5-426a-aec0-d919470e0cd6@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Jairo Bochi <jairo.bochi@xxxxxxxxx> wrote:

I'm sorry. Counterexample to what?

On Apr 8, 2:30 pm, Jan de Vries <my-n...@xxxxxxxxxxx> wrote:
The following result is well-known: if X is a complete metric space or a
locally compact space and f: X -> X is a continuous mapping such that
all points of X are non-wandering under f then the set of all recurrent
points (a.k. Poisson stable points in Birkhoff's terminilogy) is dense
in X. (Of course, Baire's theorem is involved in the proofs.)

Are there counter examples known in the literature? Presently, it is
quite difficult for me to regularly visit a good library, but references
would be helpful.
(I have a counter example in mind, but that is so complicated that I
don't know how to write it down, so a refrence to an already known
example would be much easier.)

Sorry for being not sufficiently clear.

I want (a reference to) an example of a Hausdorff space X and a
continuous mapping f of X into itself such that all points of X are
non-wandering under f and the set of recurrent (= Poisson stable) points
is not dense (or even empty).

It is known that such a space cannot be locally compact or completely
metrizable.

.

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