Expansive

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.

Definition

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 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).

If

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 .