, 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
is a metric space
, a homeomorphism
is said to be expansive
if there is a constant
called the expansivity constant, such that for any 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
an expansive homeomorphism, the theorem of uniform expansivity states that for every
there is an
such that for each pair
of points of
, there is an
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