If (X,d) is a metric space, a homeomorphism f:XX is said to be expansive if there is a constant ε0>0, called the expansivity constant, such that for any two points of X, their n-th iterates are at least ε0 apart for some integer n; i.e. if for any pair of points xy in X there is n such that d(fn(x),fn(y))ε0.

The space X is often assumed to be compactPlanetmathPlanetmath, since under that assumptionPlanetmathPlanetmath 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:XX is a continuous map, we say that X is positively expansive (or forward expansive) if there is ε0 such that, for any xy in X, there is n such that d(fn(x),fn(y))ε0.

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