Expansive homeomorphism

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 $f\colon X\to X$ is said to be expansive if there is a constant

$\varepsilon _{0}>0,$

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

$d(f^{n}(x),f^{n}(y))\geq \varepsilon _{0}$.

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 $d'$ is any other metric generating the same topology as $d$, and if $f$ is expansive in $(X,d)$, then $f$ is expansive in $(X,d')$ (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 a

$\varepsilon _{0}$

such that, for any $x\neq y$ in $X$, there is an $n\in \mathbb {N}$ 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 $n\in \mathbb {Z}$ with $\vert n\vert \leq N$ such that

$d(f^{n}(x),f^{n}(y))>c-\delta ,$

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).

External links

• Expansive dynamical systems on scholarpedia

expansive at PlanetMath. uniform expansivity at PlanetMath.