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# expansive

If $(X,d)$ is a metric space, a homeomorphism $f\colon X\to X$ is said to be expansive if there is a constant $\varepsilon_{0}>0$, called the expansivity constant, such that for any two points of $X$, their $n$-th iterates are at least $\varepsilon_{0}$ apart for some integer $n$; i.e. if for any pair of points $x\neq y$ in $X$ there is $n\in\mathbb{Z}$ such that $d(f^{n}(x),f^{n}(y))\geq\varepsilon_{0}$.

The space $X$ is often assumed to be compact, since under that assumption expansivity is a topological property; i.e. any map which is topologically conjugate to $f$ is expansive if $f$ is expansive (possibly with a different expansivity constant).

If $f\colon X\to X$ is a continuous map, we say that $X$ is positively expansive (or forward expansive) if there is $\varepsilon_{0}$ such that, for any $x\neq y$ in $X$, there is $n\in\mathbb{N}$ such that $d(f^{n}(x),f^{n}(y))\geq\varepsilon_{0}$.

Remarks. The latter condition is much stronger than expansivity. In fact, one can prove that if $X$ is compact and $f$ is a positively expansive homeomorphism, then $X$ is finite (proof).

## Mathematics Subject Classification

37B99*no label found*

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## Comments

## Regarding the definition of (forward exansivity)

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.

## Re: Regarding the definition of (forward exansivity)

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).