If is a metric space, a homeomorphism is said to be expansive if there is a constant , called the expansivity constant, such that for any two points of , their -th iterates are at least apart for some integer ; i.e. if for any pair of points in there is such that .
The space is often assumed to be compact, since under that assumption expansivity is a topological property; i.e. any map which is topologically conjugate to is expansive if is expansive (possibly with a different expansivity constant).
If is a continuous map, we say that is positively expansive (or forward expansive) if there is such that, for any in , there is such that .
Remarks. The latter condition is much stronger than expansivity. In fact, one can prove that if is compact and is a positively expansive homeomorphism, then is finite (proof (http://planetmath.org/OnlyCompactMetricSpacesThatAdmitAPostivelyExpansiveHomeomorphismAreDiscreteSpaces)).
|Date of creation||2013-03-22 13:47:48|
|Last modified on||2013-03-22 13:47:48|
|Last modified by||Koro (127)|