Expansive homeomorphism

If ${\displaystyle (X,d)}$ is a metric space, a homeomorphism ${\displaystyle f\colon X\to X}$ is said to be expansive if there is a constant

${\displaystyle \varepsilon _{0}>0,}$

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

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

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

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

If

${\displaystyle f\colon X\to X}$

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

${\displaystyle \varepsilon _{0}}$

such that, for any ${\displaystyle x\neq y}$ in ${\displaystyle X}$, there is an ${\displaystyle n\in \mathbb {N} }$ such that ${\displaystyle d(f^{n}(x),f^{n}(y))\geq \varepsilon _{0}}$.

Given f an expansive homeomorphism of a compact metric space, the theorem of uniform expansivity states that for every ${\displaystyle \epsilon >0}$ and ${\displaystyle \delta >0}$ there is an ${\displaystyle N>0}$ such that for each pair ${\displaystyle x,y}$ of points of ${\displaystyle X}$ such that ${\displaystyle d(x,y)>\epsilon }$, there is an ${\displaystyle n\in \mathbb {Z} }$ with ${\displaystyle \vert n\vert \leq N}$ such that

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

where ${\displaystyle c}$ is the expansivity constant of ${\displaystyle f}$ (proof).

Positive expansivity is much stronger than expansivity. In fact, one can prove that if ${\displaystyle X}$ is compact and ${\displaystyle f}$ is a positively expansive homeomorphism, then ${\displaystyle X}$ is finite (proof).

• Expansive dynamical systems on scholarpedia

This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike License: expansive, uniform expansivity.