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.
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
such that Note that in this definition,
can be positive or negative, and so
may be expansive in the forward or backward directions.
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
(possibly with a different expansivity constant).
If is a continuous map, we say that
is positively expansive (or forward expansive) if there is a such that, for any
ε
Given f an expansive homeomorphism of a compact metric space, the theorem of uniform expansivity states that for every
ϵ > 0
such that for each pair
is the expansivity constant of
Positive expansivity is much stronger than expansivity.
In fact, one can prove that if
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.