Definitions

# Expansive

[ik-span-siv]
In mathematics, the notion of expansivity formalizes the notion of points moving away from one-another under the action of an iterated function. The idea of expansivity is fairly rigid, as the definition of positive expansivity, below, as well as the Schwarz-Ahlfors-Pick theorem demonstrate.

## Definition

If $\left(X,d\right)$ is a metric space, a homeomorphism $fcolon Xto X$ is said to be expansive if there is a constant

$varepsilon_0>0,$

called the expansivity constant, such that for any pair of points $xneq y$ in $X$ there is an integer $n$ such that

$d\left(f^n\left(x\right),f^n\left(y\right)\right)geqvarepsilon_0$.

Note that in this definition, $n$ can be positive or negative, and so $f$ may be expansive in the forward or backward directions.

The space $X$ is often assumed to be compact, since under that assumption expansivity is a topological property; i.e. if $d\text{'}$ is any other metric generating the same topology as $d$, and if $f$ is expansive in $\left(X,d\right)$, then $f$ is expansive in $\left(X,d\text{'}\right)$ (possibly with a different expansivity constant).

If

$fcolon Xto X$

is a continuous map, we say that $X$ is positively expansive (or forward expansive) if there is a

$varepsilon_0$

such that, for any $xneq y$ in $X$, there is an $ninmathbb\left\{N\right\}$ such that $d\left(f^n\left(x\right),f^n\left(y\right)\right)geq varepsilon_0$.

## Theorem of uniform expansivity

Given f an expansive homeomorphism, the theorem of uniform expansivity states that for every $epsilon>0$ and $delta>0$ there is an $N>0$ such that for each pair $x,y$ of points of $X$ such that $d\left(x,y\right)>epsilon$, there is an $nin mathbb\left\{Z\right\}$ with $vert nvertleq N$ such that

$d\left(f^n\left(x\right),f^n\left(y\right)\right) > c-delta,$

where $c$ is the expansivity constant of $f$ (proof).

## Discussion

Positive expansivity 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).

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