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.
If is a metric space, a homeomorphism is said to be expansive if there is a constant
called the expansivity constant, such that for every pair of points in there is an integer such that
Note that in this definition, can be positive or negative, and so may be expansive in the forward or backward directions.
The space is often assumed to be compact, since under that assumption expansivity is a topological property; i.e. if is any other metric generating the same topology as , and if is expansive in , then is expansive in (possibly with a different expansivity constant).
is a continuous map, we say that is positively expansive (or forward expansive) if there is a
such that, for any in , there is an such that .
Theorem of uniform expansivity
Given f an expansive homeomorphism of a compact metric space, the theorem of uniform expansivity states that for every and there is an such that for each pair of points of such that , there is an with such that
where is the expansivity constant of (proof).
Positive expansivity is much stronger than expansivity. In fact, one can prove that if is compact and is a positively
expansive homeomorphism, then is finite (proof).
This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike License: expansive, uniform expansivity.