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expansivity, positively expansive, forward expansive
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I think that you are missing a few details , such as if f is expansive
with regards to (X,d) then there is e
such that for every x != y there is n where d(f^n(x),f^n(y)>e.
Is this n universal or is n depends on x and y ?

if m>n does it mean that d(f^m(x),f^m(y))>e ?

If you dont demand either and you dont demand that f is continues
then there is an example where f is forward expansive on (X,d)
X compact and not finite.

The example is X cantor set d-the standard euclid metric.
We can consider X as 2^omega (infinite series of zeros or ones)
and the (forward) expansive map is the "shift left" map ,
a well known mixing map.

That's right; there was an error. In my last comment "map" should have been "homeomorphism". The idea is that if f is an homeomorphism then forward expansivity is much stronger than expansivity (if f is not homeo then there is no point in comparing them, since "expansivity" alone is not defined if f is not an homeo).

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