Hi 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.
On Sat, 31 Oct 2009 12:00:02 +0000 (GMT), rabeb.hattab
<rabeb.hat...@gmail.com> wrote: >Hi >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.
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.)
> 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
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