expansive

If $(X,d)$ is a metric space, a homeomorphism $f:X\to X$ is said to be expansive if there is a constant ${\epsilon }_0>0$, called the expansivity constant, such that for any two points of $X$, their $n$-th iterates are at least ${\epsilon }_0$ apart for some integer $n$; i.e. if for any pair of points $x\ne y$ in $X$ there is $n\in \mathbb{Z}$ such that $d(f^n(x),f^n(y))\ge {\epsilon }_0$.

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 $f:X\to X$ is a continuous map, we say that $X$ is positively expansive (or forward expansive) if there is ${\epsilon }_0$ such that, for any $x\ne y$ in $X$, there is $n\in \mathbb{N}$ such that $d(f^n(x),f^n(y))\ge {\epsilon }_0$.

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 (http://planetmath.org/OnlyCompactMetricSpacesThatAdmitAPostivelyExpansiveHomeomorphismAreDiscreteSpaces)).

Title	expansive
Canonical name	Expansive
Date of creation	2013-03-22 13:47:48
Last modified on	2013-03-22 13:47:48
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	14
Author	Koro (127)
Entry type	Definition
Classification	msc 37B99
Defines	expansivity
Defines	positively expansive
Defines	forward expansive