Expansive

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 $(X,d)$ is a metric space, a homeomorphism $fcolon\; X\; o\; X$ is said to be expansive if there is a constant

:$varepsilon\_0>0,$

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

:$d(f^n(x),f^n(y))geqvarepsilon\_0$.

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 thatassumption 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 $(X,d)$, then $f$ is expansive in $(X,d\text{'})$ (possibly with a different expansivity constant).

If

:$fcolon\; X\; o\; X$

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

:$varepsilon\_0$

such that, for any $x\; eq\; y$ in $X$, there is an $ninmathbb\{N\}$ such that $d(f^n(x),f^n(y))geq\; varepsilon\_0$.

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 $d(x,y)>epsilon$, there is an $nin\; mathbb\{Z\}$ with $vert\; nvertleq\; N$ such that

:$d(f^n(x),f^n(y))\; >\; c-delta,$

where $c$ is the expansivity constant of $f$ ( [http://planetmath.org/?op=getobj&from=objects&id=4678 proof] ).

Discussion

Positive expansivity is much stronger than expansivity. In fact, one can prove that if $X$ is compact and $f$ is a positivelyexpansive homeomorphism, then $X$ is finite ( [http://planetmath.org/?op=getobj&from=objects&id=4677 proof] ).