1 able or tending to expand or characterized by expansion; "Expansive materials"; "the expansive force of fire" [ant: unexpansive]
2 impressive in scale; "an expansive lifestyle"; "in the grand manner" [syn: grand]
3 marked by exaggerated feelings of euphoria and delusions of grandeur
4 friendly and open and willing to talk; "wine made the guest expansive" [syn: talkative]
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.
DefinitionIf (X,d) is a metric space, a homeomorphism f\colon X\to X is said to be expansive if there is a constant
called the expansivity constant, such that for any two points of X, their n-th iterates are at least \varepsilon_0 apart for some integer n; i.e. if for any pair of points x\neq y in X there is n\in\mathbb such that
Note that in this definition, n can be positive or negative, and so f may be expansive in the forward or backward directions.
The space X is often assumed to be compact, since under that assumption expansivity is a topological property; i.e. if d' is any other metric generating the same topology as d, and if f is expansive in (X,d), then f is expansive in (X,d') (possibly with a different expansivity constant).
- f\colon X\to X
is a continuous map, we say that X is positively expansive (or forward expansive) if there is a
such that, for any x\neq y in X, there is an n\in\mathbb such that d(f^n(x),f^n(y))\geq \varepsilon_0.
Theorem of uniform expansivityGiven f an expansive homeomorphism, the theorem of uniform expansivity states that for every \epsilon>0 and \delta>0 there is an N>0 such that for each pair x,y of points of X such that d(x,y)>\epsilon, there is an n\in \mathbb with \vert n\vert\leq N such that
- d(f^n(x),f^n(y)) > c-\delta,
where c is the expansivity constant of f (proof).
DiscussionPositive expansivity is much stronger than expansivity. In fact, one can prove that if X is compact and f is a positively expansive homeomorphism, then X is finite (proof).
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