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

is a metric space, a homeomorphism is said to be expansive if there is a constant

called the expansivity constant such that

may be expansive in the forward or backward directions.

is often assumed to be compact, since under that assumption expansivity is a topological property (possibly with a different expansivity constant).

If

is positively expansive (or forward expansive) if there is a

.

Theorem of uniform expansivity

Given f such that

(http://planetmath.org/?op=getobj).

Discussion

is finite (http://planetmath.org/?op=getobj).

External links

• Expansive dynamical systems on scholarpedia