recurerent map
rabeb.hattab
A continuous map $f$ is recurrent if for all $a>0$ there exists an
integer $n$ such that $d(f^n,id)<a$.
It is say to be equicontinuous if ${f,f^2,.....f^n,...} $ is
equicontinuous.
Is there an implication between the two notions? If not is there
example showing that none of the two implication is true.
David C. Ullrich
Regardless of exactly what your definition of "recurrent" means,
it seems pretty clear that the map f : R -> R defined by f(x) = x + 1
is equicontinuous but not recurrent. Or f(x) = 0, for that matter.
Since counterexamples are so obvious one suspects you had
some unstated assumptions in mind. (Also, exactly what does
the "d" mean in the definition?)
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Olivier
[...]
> Since counterexamples are so obvious one suspects you had
> some unstated assumptions in mind. (Also, exactly what does
> the "d" mean in the definition?)
Something like on a compact metrizable space :)
In such a case, if f is recurrent it is
equicontinuous. I suspect the converse
to be true when f is onto and false
otherwise (f(x)=x/2 on [0,1] is
a counter example).
Best,
Amities,
Olivier