Expansive homeomorphism

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

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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).

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  .

Theorem of uniform expansivity

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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).

Discussion

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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).

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This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike License: expansive, uniform expansivity.